LMFDB - Embedded newform 320.6.d.b.161.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,6,Mod(161,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("320.161");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	320.d (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:	$51.3228223402$
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} - 4x^{7} - 384x^{6} + 506x^{5} + 49869x^{4} + 29654x^{3} - 2235516x^{2} - 1528906x + 34180205 $ x^8 - 4x^7 - 384x^6 + 506x^5 + 49869x^4 + 29654x^3 - 2235516x^2 - 1528906*x + 34180205
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.8
Root			$14.2856 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	320.161
Dual form			320.6.d.b.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.5713i q^{3} -25.0000i q^{5} +220.359 q^{7} -517.174 q^{9} +O(q^{10})$	$q+27.5713i q^{3} -25.0000i q^{5} +220.359 q^{7} -517.174 q^{9} -64.4115i q^{11} -703.006i q^{13} +689.281 q^{15} -207.573 q^{17} +2790.91i q^{19} +6075.56i q^{21} +3258.47 q^{23} -625.000 q^{25} -7559.32i q^{27} +7896.13i q^{29} -5051.23 q^{31} +1775.91 q^{33} -5508.96i q^{35} +4915.89i q^{37} +19382.8 q^{39} +7036.16 q^{41} +15147.0i q^{43} +12929.4i q^{45} +9305.48 q^{47} +31750.9 q^{49} -5723.05i q^{51} +4938.18i q^{53} -1610.29 q^{55} -76949.0 q^{57} +38441.5i q^{59} -27779.9i q^{61} -113964. q^{63} -17575.2 q^{65} -7780.18i q^{67} +89840.2i q^{69} -60101.8 q^{71} +36397.3 q^{73} -17232.0i q^{75} -14193.6i q^{77} -36768.7 q^{79} +82746.7 q^{81} -21132.9i q^{83} +5189.32i q^{85} -217706. q^{87} -63887.8 q^{89} -154913. i q^{91} -139269. i q^{93} +69772.9 q^{95} -14388.5 q^{97} +33312.0i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 320 q^{7} - 1192 q^{9}+O(q^{10}) $ 8 * q + 320 * q^7 - 1192 * q^9	$ 8 q + 320 q^{7} - 1192 q^{9} - 2400 q^{17} - 1760 q^{23} - 5000 q^{25} - 31040 q^{31} + 3760 q^{33} - 4480 q^{39} + 21584 q^{41} - 47680 q^{47} + 82824 q^{49} - 18400 q^{55} - 106640 q^{57} - 322400 q^{63} - 44000 q^{65} - 246720 q^{71} + 46400 q^{73} - 325760 q^{79} - 82328 q^{81} - 636320 q^{87} - 78192 q^{89} + 40800 q^{95} + 24960 q^{97}+O(q^{100}) $ 8 * q + 320 * q^7 - 1192 * q^9 - 2400 * q^17 - 1760 * q^23 - 5000 * q^25 - 31040 * q^31 + 3760 * q^33 - 4480 * q^39 + 21584 * q^41 - 47680 * q^47 + 82824 * q^49 - 18400 * q^55 - 106640 * q^57 - 322400 * q^63 - 44000 * q^65 - 246720 * q^71 + 46400 * q^73 - 325760 * q^79 - 82328 * q^81 - 636320 * q^87 - 78192 * q^89 + 40800 * q^95 + 24960 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$.

$n$	$191$	$257$	$261$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	27.5713i	1.76870i	0.466828	+	0.884348i	$0.345397\pi$
$3$	27.5713i	1.76870i	−0.466828	+	0.884348i	$0.654603\pi$
$4$	0	0
$5$	− 25.0000i	− 0.447214i
$5$	− 25.0000i	− 0.447214i
$6$	0	0
$7$	220.359	1.69975	0.849875	−	0.526985i	$-0.176678\pi$
$7$	220.359	1.69975	0.849875	+	0.526985i	$0.176678\pi$
$8$	0	0
$9$	−517.174	−2.12829
$10$	0	0
$11$	− 64.4115i	− 0.160503i	−0.996775	−	0.0802513i	$-0.974428\pi$
$11$	− 64.4115i	− 0.160503i	0.996775	−	0.0802513i	$-0.0255723\pi$
$12$	0	0
$13$	− 703.006i	− 1.15372i	−0.816843	−	0.576860i	$-0.804278\pi$
$13$	− 703.006i	− 1.15372i	0.816843	−	0.576860i	$-0.195722\pi$
$14$	0	0
$15$	689.281	0.790985
$16$	0	0
$17$	−207.573	−0.174200	−0.0871001	−	0.996200i	$-0.527760\pi$
$17$	−207.573	−0.174200	−0.0871001	+	0.996200i	$0.527760\pi$
$18$	0	0
$19$	2790.91i	1.77363i	0.462127	+	0.886814i	$0.347087\pi$
$19$	2790.91i	1.77363i	−0.462127	+	0.886814i	$0.652913\pi$
$20$	0	0
$21$	6075.56i	3.00634i
$22$	0	0
$23$	3258.47	1.28438	0.642192	−	0.766544i	$-0.278025\pi$
$23$	3258.47	1.28438	0.642192	+	0.766544i	$0.278025\pi$
$24$	0	0
$25$	−625.000	−0.200000
$26$	0	0
$27$	− 7559.32i	− 1.99560i
$28$	0	0
$29$	7896.13i	1.74349i	0.489961	+	0.871745i	$0.337011\pi$
$29$	7896.13i	1.74349i	−0.489961	+	0.871745i	$0.662989\pi$
$30$	0	0
$31$	−5051.23	−0.944046	−0.472023	−	0.881586i	$-0.656476\pi$
$31$	−5051.23	−0.944046	−0.472023	+	0.881586i	$0.656476\pi$
$32$	0	0
$33$	1775.91	0.283880
$34$	0	0
$35$	− 5508.96i	− 0.760151i
$36$	0	0
$37$	4915.89i	0.590334i	0.955446	+	0.295167i	$0.0953753\pi$
$37$	4915.89i	0.590334i	−0.955446	+	0.295167i	$0.904625\pi$
$38$	0	0
$39$	19382.8	2.04058
$40$	0	0
$41$	7036.16	0.653696	0.326848	−	0.945077i	$-0.394013\pi$
$41$	7036.16	0.653696	0.326848	+	0.945077i	$0.394013\pi$
$42$	0	0
$43$	15147.0i	1.24927i	0.780918	+	0.624633i	$0.214752\pi$
$43$	15147.0i	1.24927i	−0.780918	+	0.624633i	$0.785248\pi$
$44$	0	0
$45$	12929.4i	0.951799i
$46$	0	0
$47$	9305.48	0.614461	0.307230	−	0.951635i	$-0.400598\pi$
$47$	9305.48	0.614461	0.307230	+	0.951635i	$0.400598\pi$
$48$	0	0
$49$	31750.9	1.88915
$50$	0	0
$51$	− 5723.05i	− 0.308107i
$52$	0	0
$53$	4938.18i	0.241478i	0.992684	+	0.120739i	$0.0385264\pi$
$53$	4938.18i	0.241478i	−0.992684	+	0.120739i	$0.961474\pi$
$54$	0	0
$55$	−1610.29	−0.0717789
$56$	0	0
$57$	−76949.0	−3.13701
$58$	0	0
$59$	38441.5i	1.43771i	0.695162	+	0.718853i	$0.255333\pi$
$59$	38441.5i	1.43771i	−0.695162	+	0.718853i	$0.744667\pi$
$60$	0	0
$61$	− 27779.9i	− 0.955886i	−0.878391	−	0.477943i	$-0.841382\pi$
$61$	− 27779.9i	− 0.955886i	0.878391	−	0.477943i	$-0.158618\pi$
$62$	0	0
$63$	−113964.	−3.61756
$64$	0	0
$65$	−17575.2	−0.515960
$66$	0	0
$67$	− 7780.18i	− 0.211740i	−0.994380	−	0.105870i	$-0.966237\pi$
$67$	− 7780.18i	− 0.211740i	0.994380	−	0.105870i	$-0.0337627\pi$
$68$	0	0
$69$	89840.2i	2.27168i
$70$	0	0
$71$	−60101.8	−1.41495	−0.707476	−	0.706737i	$-0.750166\pi$
$71$	−60101.8	−1.41495	−0.707476	+	0.706737i	$0.750166\pi$
$72$	0	0
$73$	36397.3	0.799396	0.399698	−	0.916647i	$-0.369115\pi$
$73$	36397.3	0.799396	0.399698	+	0.916647i	$0.369115\pi$
$74$	0	0
$75$	− 17232.0i	− 0.353739i
$76$	0	0
$77$	− 14193.6i	− 0.272814i
$78$	0	0
$79$	−36768.7	−0.662843	−0.331421	−	0.943483i	$-0.607528\pi$
$79$	−36768.7	−0.662843	−0.331421	+	0.943483i	$0.607528\pi$
$80$	0	0
$81$	82746.7	1.40132
$82$	0	0
$83$	− 21132.9i	− 0.336716i	−0.985726	−	0.168358i	$-0.946153\pi$
$83$	− 21132.9i	− 0.336716i	0.985726	−	0.168358i	$-0.0538465\pi$
$84$	0	0
$85$	5189.32i	0.0779047i
$86$	0	0
$87$	−217706.	−3.08370
$88$	0	0
$89$	−63887.8	−0.854955	−0.427477	−	0.904026i	$-0.640598\pi$
$89$	−63887.8	−0.854955	−0.427477	+	0.904026i	$0.640598\pi$
$90$	0	0
$91$	− 154913.i	− 1.96104i
$92$	0	0
$93$	− 139269.i	− 1.66973i
$94$	0	0
$95$	69772.9	0.793190
$96$	0	0
$97$	−14388.5	−0.155269	−0.0776346	−	0.996982i	$-0.524737\pi$
$97$	−14388.5	−0.155269	−0.0776346	+	0.996982i	$0.524737\pi$
$98$	0	0
$99$	33312.0i	0.341596i
$100$	0	0
$101$	73761.5i	0.719493i	0.933050	+	0.359747i	$0.117137\pi$
$101$	73761.5i	0.719493i	−0.933050	+	0.359747i	$0.882863\pi$
$102$	0	0
$103$	13374.5	0.124218	0.0621090	−	0.998069i	$-0.480217\pi$
$103$	13374.5	0.124218	0.0621090	+	0.998069i	$0.480217\pi$
$104$	0	0
$105$	151889.	1.34448
$106$	0	0
$107$	206058.i	1.73992i	0.493120	+	0.869961i	$0.335856\pi$
$107$	206058.i	1.73992i	−0.493120	+	0.869961i	$0.664144\pi$
$108$	0	0
$109$	73086.1i	0.589208i	0.955619	+	0.294604i	$0.0951877\pi$
$109$	73086.1i	0.589208i	−0.955619	+	0.294604i	$0.904812\pi$
$110$	0	0
$111$	−135537.	−1.04412
$112$	0	0
$113$	−154909.	−1.14125	−0.570626	−	0.821210i	$-0.693299\pi$
$113$	−154909.	−1.14125	−0.570626	+	0.821210i	$0.693299\pi$
$114$	0	0
$115$	− 81461.8i	− 0.574394i
$116$	0	0
$117$	363577.i	2.45545i
$118$	0	0
$119$	−45740.5	−0.296097
$120$	0	0
$121$	156902.	0.974239
$122$	0	0
$123$	193996.i	1.15619i
$124$	0	0
$125$	15625.0i	0.0894427i
$126$	0	0
$127$	156780.	0.862545	0.431273	−	0.902222i	$-0.358065\pi$
$127$	156780.	0.862545	0.431273	+	0.902222i	$0.358065\pi$
$128$	0	0
$129$	−417621.	−2.20957
$130$	0	0
$131$	− 241597.i	− 1.23002i	−0.788518	−	0.615012i	$-0.789151\pi$
$131$	− 241597.i	− 1.23002i	0.788518	−	0.615012i	$-0.210849\pi$
$132$	0	0
$133$	615002.i	3.01472i
$134$	0	0
$135$	−188983.	−0.892459
$136$	0	0
$137$	−34718.4	−0.158037	−0.0790184	−	0.996873i	$-0.525179\pi$
$137$	−34718.4	−0.158037	−0.0790184	+	0.996873i	$0.525179\pi$
$138$	0	0
$139$	− 148522.i	− 0.652008i	−0.945368	−	0.326004i	$-0.894298\pi$
$139$	− 148522.i	− 0.652008i	0.945368	−	0.326004i	$-0.105702\pi$
$140$	0	0
$141$	256564.i	1.08680i
$142$	0	0
$143$	−45281.7	−0.185175
$144$	0	0
$145$	197403.	0.779712
$146$	0	0
$147$	875412.i	3.34133i
$148$	0	0
$149$	95567.9i	0.352652i	0.984332	+	0.176326i	$0.0564213\pi$
$149$	95567.9i	0.352652i	−0.984332	+	0.176326i	$0.943579\pi$
$150$	0	0
$151$	445946.	1.59162	0.795810	−	0.605546i	$-0.207045\pi$
$151$	445946.	1.59162	0.795810	+	0.605546i	$0.207045\pi$
$152$	0	0
$153$	107351.	0.370748
$154$	0	0
$155$	126281.i	0.422190i
$156$	0	0
$157$	− 534503.i	− 1.73062i	−0.501240	−	0.865308i	$-0.667123\pi$
$157$	− 534503.i	− 1.73062i	0.501240	−	0.865308i	$-0.332877\pi$
$158$	0	0
$159$	−136152.	−0.427101
$160$	0	0
$161$	718033.	2.18313
$162$	0	0
$163$	− 509438.i	− 1.50184i	−0.660396	−	0.750918i	$-0.729611\pi$
$163$	− 509438.i	− 1.50184i	0.660396	−	0.750918i	$-0.270389\pi$
$164$	0	0
$165$	− 44397.7i	− 0.126955i
$166$	0	0
$167$	658419.	1.82689	0.913443	−	0.406967i	$-0.133414\pi$
$167$	658419.	1.82689	0.913443	+	0.406967i	$0.133414\pi$
$168$	0	0
$169$	−122925.	−0.331072
$170$	0	0
$171$	− 1.44339e6i	− 3.77479i
$172$	0	0
$173$	233004.i	0.591900i	0.955203	+	0.295950i	$0.0956362\pi$
$173$	233004.i	0.591900i	−0.955203	+	0.295950i	$0.904364\pi$
$174$	0	0
$175$	−137724.	−0.339950
$176$	0	0
$177$	−1.05988e6	−2.54287
$178$	0	0
$179$	300920.i	0.701970i	0.936381	+	0.350985i	$0.114153\pi$
$179$	300920.i	0.701970i	−0.936381	+	0.350985i	$0.885847\pi$
$180$	0	0
$181$	− 54124.5i	− 0.122800i	−0.998113	−	0.0613998i	$-0.980444\pi$
$181$	− 54124.5i	− 0.122800i	0.998113	−	0.0613998i	$-0.0195565\pi$
$182$	0	0
$183$	765927.	1.69067
$184$	0	0
$185$	122897.	0.264006
$186$	0	0
$187$	13370.1i	0.0279596i
$188$	0	0
$189$	− 1.66576e6i	− 3.39202i
$190$	0	0
$191$	−417470.	−0.828021	−0.414011	−	0.910272i	$-0.635872\pi$
$191$	−417470.	−0.828021	−0.414011	+	0.910272i	$0.635872\pi$
$192$	0	0
$193$	430941.	0.832769	0.416384	−	0.909189i	$-0.363297\pi$
$193$	430941.	0.832769	0.416384	+	0.909189i	$0.363297\pi$
$194$	0	0
$195$	− 484569.i	− 0.912576i
$196$	0	0
$197$	289267.i	0.531047i	0.964104	+	0.265523i	$0.0855448\pi$
$197$	289267.i	0.531047i	−0.964104	+	0.265523i	$0.914455\pi$
$198$	0	0
$199$	−812396.	−1.45424	−0.727119	−	0.686512i	$-0.759141\pi$
$199$	−812396.	−1.45424	−0.727119	+	0.686512i	$0.759141\pi$
$200$	0	0
$201$	214509.	0.374504
$202$	0	0
$203$	1.73998e6i	2.96349i
$204$	0	0
$205$	− 175904.i	− 0.292342i
$206$	0	0
$207$	−1.68520e6	−2.73354
$208$	0	0
$209$	179767.	0.284672
$210$	0	0
$211$	227948.i	0.352476i	0.984348	+	0.176238i	$0.0563928\pi$
$211$	227948.i	0.352476i	−0.984348	+	0.176238i	$0.943607\pi$
$212$	0	0
$213$	− 1.65708e6i	− 2.50262i
$214$	0	0
$215$	378675.	0.558689
$216$	0	0
$217$	−1.11308e6	−1.60464
$218$	0	0
$219$	1.00352e6i	1.41389i
$220$	0	0
$221$	145925.i	0.200978i
$222$	0	0
$223$	371367.	0.500082	0.250041	−	0.968235i	$-0.419556\pi$
$223$	371367.	0.500082	0.250041	+	0.968235i	$0.419556\pi$
$224$	0	0
$225$	323234.	0.425658
$226$	0	0
$227$	− 761652.i	− 0.981052i	−0.871427	−	0.490526i	$-0.836805\pi$
$227$	− 761652.i	− 0.981052i	0.871427	−	0.490526i	$-0.163195\pi$
$228$	0	0
$229$	325400.i	0.410043i	0.978757	+	0.205021i	$0.0657264\pi$
$229$	325400.i	0.410043i	−0.978757	+	0.205021i	$0.934274\pi$
$230$	0	0
$231$	391336.	0.482525
$232$	0	0
$233$	−527689.	−0.636778	−0.318389	−	0.947960i	$-0.603142\pi$
$233$	−527689.	−0.636778	−0.318389	+	0.947960i	$0.603142\pi$
$234$	0	0
$235$	− 232637.i	− 0.274795i
$236$	0	0
$237$	− 1.01376e6i	− 1.17237i
$238$	0	0
$239$	1.09227e6	1.23691	0.618454	−	0.785821i	$-0.287759\pi$
$239$	1.09227e6	1.23691	0.618454	+	0.785821i	$0.287759\pi$
$240$	0	0
$241$	−63366.4	−0.0702776	−0.0351388	−	0.999382i	$-0.511187\pi$
$241$	−63366.4	−0.0702776	−0.0351388	+	0.999382i	$0.511187\pi$
$242$	0	0
$243$	444515.i	0.482915i
$244$	0	0
$245$	− 793772.i	− 0.844852i
$246$	0	0
$247$	1.96203e6	2.04627
$248$	0	0
$249$	582661.	0.595549
$250$	0	0
$251$	− 1.59819e6i	− 1.60120i	−0.599201	−	0.800599i	$-0.704515\pi$
$251$	− 1.59819e6i	− 1.60120i	0.599201	−	0.800599i	$-0.295485\pi$
$252$	0	0
$253$	− 209883.i	− 0.206147i
$254$	0	0
$255$	−143076.	−0.137790
$256$	0	0
$257$	−478358.	−0.451773	−0.225886	−	0.974154i	$-0.572528\pi$
$257$	−478358.	−0.451773	−0.225886	+	0.974154i	$0.572528\pi$
$258$	0	0
$259$	1.08326e6i	1.00342i
$260$	0	0
$261$	− 4.08367e6i	− 3.71065i
$262$	0	0
$263$	−720034.	−0.641895	−0.320947	−	0.947097i	$-0.604001\pi$
$263$	−720034.	−0.641895	−0.320947	+	0.947097i	$0.604001\pi$
$264$	0	0
$265$	123454.	0.107992
$266$	0	0
$267$	− 1.76147e6i	− 1.51216i
$268$	0	0
$269$	1.74800e6i	1.47286i	0.676514	+	0.736430i	$0.263490\pi$
$269$	1.74800e6i	1.47286i	−0.676514	+	0.736430i	$0.736510\pi$
$270$	0	0
$271$	1.80305e6	1.49137	0.745683	−	0.666300i	$-0.232123\pi$
$271$	1.80305e6	1.49137	0.745683	+	0.666300i	$0.232123\pi$
$272$	0	0
$273$	4.27116e6	3.46848
$274$	0	0
$275$	40257.2i	0.0321005i
$276$	0	0
$277$	243112.i	0.190374i	0.995459	+	0.0951869i	$0.0303449\pi$
$277$	243112.i	0.190374i	−0.995459	+	0.0951869i	$0.969655\pi$
$278$	0	0
$279$	2.61237e6	2.00920
$280$	0	0
$281$	−377345.	−0.285084	−0.142542	−	0.989789i	$-0.545528\pi$
$281$	−377345.	−0.285084	−0.142542	+	0.989789i	$0.545528\pi$
$282$	0	0
$283$	904709.i	0.671495i	0.941952	+	0.335748i	$0.108989\pi$
$283$	904709.i	0.671495i	−0.941952	+	0.335748i	$0.891011\pi$
$284$	0	0
$285$	1.92373e6i	1.40291i
$286$	0	0
$287$	1.55048e6	1.11112
$288$	0	0
$289$	−1.37677e6	−0.969654
$290$	0	0
$291$	− 396708.i	− 0.274624i
$292$	0	0
$293$	1.57254e6i	1.07012i	0.844814	+	0.535060i	$0.179711\pi$
$293$	1.57254e6i	1.07012i	−0.844814	+	0.535060i	$0.820289\pi$
$294$	0	0
$295$	961037.	0.642962
$296$	0	0
$297$	−486907.	−0.320299
$298$	0	0
$299$	− 2.29073e6i	− 1.48182i
$300$	0	0
$301$	3.33777e6i	2.12344i
$302$	0	0
$303$	−2.03370e6	−1.27256
$304$	0	0
$305$	−694498.	−0.427485
$306$	0	0
$307$	1.50494e6i	0.911326i	0.890152	+	0.455663i	$0.150598\pi$
$307$	1.50494e6i	0.911326i	−0.890152	+	0.455663i	$0.849402\pi$
$308$	0	0
$309$	368752.i	0.219704i
$310$	0	0
$311$	50915.6	0.0298504	0.0149252	−	0.999889i	$-0.495249\pi$
$311$	50915.6	0.0298504	0.0149252	+	0.999889i	$0.495249\pi$
$312$	0	0
$313$	−1.05135e6	−0.606581	−0.303290	−	0.952898i	$-0.598085\pi$
$313$	−1.05135e6	−0.606581	−0.303290	+	0.952898i	$0.598085\pi$
$314$	0	0
$315$	2.84909e6i	1.61782i
$316$	0	0
$317$	591005.i	0.330326i	0.986266	+	0.165163i	$0.0528151\pi$
$317$	591005.i	0.330326i	−0.986266	+	0.165163i	$0.947185\pi$
$318$	0	0
$319$	508602.	0.279834
$320$	0	0
$321$	−5.68128e6	−3.07740
$322$	0	0
$323$	− 579318.i	− 0.308966i
$324$	0	0
$325$	439379.i	0.230744i
$326$	0	0
$327$	−2.01508e6	−1.04213
$328$	0	0
$329$	2.05054e6	1.04443
$330$	0	0
$331$	1.47751e6i	0.741242i	0.928784	+	0.370621i	$0.120855\pi$
$331$	1.47751e6i	0.741242i	−0.928784	+	0.370621i	$0.879145\pi$
$332$	0	0
$333$	− 2.54237e6i	− 1.25640i
$334$	0	0
$335$	−194505.	−0.0946930
$336$	0	0
$337$	3.32321e6	1.59398	0.796990	−	0.603993i	$-0.206425\pi$
$337$	3.32321e6	1.59398	0.796990	+	0.603993i	$0.206425\pi$
$338$	0	0
$339$	− 4.27104e6i	− 2.01853i
$340$	0	0
$341$	325357.i	0.151522i
$342$	0	0
$343$	3.29301e6	1.51133
$344$	0	0
$345$	2.24600e6	1.01593
$346$	0	0
$347$	602144.i	0.268458i	0.990950	+	0.134229i	$0.0428558\pi$
$347$	602144.i	0.268458i	−0.990950	+	0.134229i	$0.957144\pi$
$348$	0	0
$349$	1.58858e6i	0.698147i	0.937095	+	0.349073i	$0.113504\pi$
$349$	1.58858e6i	0.698147i	−0.937095	+	0.349073i	$0.886496\pi$
$350$	0	0
$351$	−5.31425e6	−2.30237
$352$	0	0
$353$	319264.	0.136368	0.0681841	−	0.997673i	$-0.478279\pi$
$353$	319264.	0.136368	0.0681841	+	0.997673i	$0.478279\pi$
$354$	0	0
$355$	1.50255e6i	0.632786i
$356$	0	0
$357$	− 1.26112e6i	− 0.523705i
$358$	0	0
$359$	3.83807e6	1.57173	0.785863	−	0.618401i	$-0.212219\pi$
$359$	3.83807e6	1.57173	0.785863	+	0.618401i	$0.212219\pi$
$360$	0	0
$361$	−5.31310e6	−2.14576
$362$	0	0
$363$	4.32599e6i	1.72313i
$364$	0	0
$365$	− 909932.i	− 0.357501i
$366$	0	0
$367$	−2.02410e6	−0.784455	−0.392227	−	0.919868i	$-0.628295\pi$
$367$	−2.02410e6	−0.784455	−0.392227	+	0.919868i	$0.628295\pi$
$368$	0	0
$369$	−3.63892e6	−1.39125
$370$	0	0
$371$	1.08817e6i	0.410451i
$372$	0	0
$373$	− 4.39562e6i	− 1.63587i	−0.575311	−	0.817935i	$-0.695119\pi$
$373$	− 4.39562e6i	− 1.63587i	0.575311	−	0.817935i	$-0.304881\pi$
$374$	0	0
$375$	−430801.	−0.158197
$376$	0	0
$377$	5.55103e6	2.01150
$378$	0	0
$379$	− 2.04075e6i	− 0.729780i	−0.931051	−	0.364890i	$-0.881107\pi$
$379$	− 2.04075e6i	− 0.729780i	0.931051	−	0.364890i	$-0.118893\pi$
$380$	0	0
$381$	4.32263e6i	1.52558i
$382$	0	0
$383$	−2.64545e6	−0.921516	−0.460758	−	0.887526i	$-0.652422\pi$
$383$	−2.64545e6	−0.921516	−0.460758	+	0.887526i	$0.652422\pi$
$384$	0	0
$385$	−354841.	−0.122006
$386$	0	0
$387$	− 7.83363e6i	− 2.65880i
$388$	0	0
$389$	− 1.90022e6i	− 0.636693i	−0.947974	−	0.318346i	$-0.896873\pi$
$389$	− 1.90022e6i	− 0.636693i	0.947974	−	0.318346i	$-0.103127\pi$
$390$	0	0
$391$	−676371.	−0.223740
$392$	0	0
$393$	6.66113e6	2.17554
$394$	0	0
$395$	919217.i	0.296432i
$396$	0	0
$397$	− 716277.i	− 0.228089i	−0.993476	−	0.114045i	$-0.963619\pi$
$397$	− 716277.i	− 0.228089i	0.993476	−	0.114045i	$-0.0363807\pi$
$398$	0	0
$399$	−1.69564e7	−5.33213
$400$	0	0
$401$	−1.77803e6	−0.552175	−0.276088	−	0.961132i	$-0.589038\pi$
$401$	−1.77803e6	−0.552175	−0.276088	+	0.961132i	$0.589038\pi$
$402$	0	0
$403$	3.55105e6i	1.08917i
$404$	0	0
$405$	− 2.06867e6i	− 0.626690i
$406$	0	0
$407$	316640.	0.0947502
$408$	0	0
$409$	3.15560e6	0.932767	0.466384	−	0.884583i	$-0.345557\pi$
$409$	3.15560e6	0.932767	0.466384	+	0.884583i	$0.345557\pi$
$410$	0	0
$411$	− 957230.i	− 0.279519i
$412$	0	0
$413$	8.47091e6i	2.44374i
$414$	0	0
$415$	−528323.	−0.150584
$416$	0	0
$417$	4.09493e6	1.15320
$418$	0	0
$419$	1.73881e6i	0.483857i	0.970294	+	0.241929i	$0.0777800\pi$
$419$	1.73881e6i	0.483857i	−0.970294	+	0.241929i	$0.922220\pi$
$420$	0	0
$421$	− 4.05279e6i	− 1.11442i	−0.830371	−	0.557210i	$-0.811872\pi$
$421$	− 4.05279e6i	− 1.11442i	0.830371	−	0.557210i	$-0.188128\pi$
$422$	0	0
$423$	−4.81255e6	−1.30775
$424$	0	0
$425$	129733.	0.0348400
$426$	0	0
$427$	− 6.12154e6i	− 1.62477i
$428$	0	0
$429$	− 1.24847e6i	− 0.327519i
$430$	0	0
$431$	−4.04038e6	−1.04768	−0.523840	−	0.851817i	$-0.675501\pi$
$431$	−4.04038e6	−1.04768	−0.523840	+	0.851817i	$0.675501\pi$
$432$	0	0
$433$	4.95477e6	1.27000	0.634999	−	0.772513i	$-0.281000\pi$
$433$	4.95477e6	1.27000	0.634999	+	0.772513i	$0.281000\pi$
$434$	0	0
$435$	5.44265e6i	1.37907i
$436$	0	0
$437$	9.09412e6i	2.27802i
$438$	0	0
$439$	2.27775e6	0.564086	0.282043	−	0.959402i	$-0.408988\pi$
$439$	2.27775e6	0.564086	0.282043	+	0.959402i	$0.408988\pi$
$440$	0	0
$441$	−1.64207e7	−4.02065
$442$	0	0
$443$	− 2.88598e6i	− 0.698688i	−0.936995	−	0.349344i	$-0.886404\pi$
$443$	− 2.88598e6i	− 0.698688i	0.936995	−	0.349344i	$-0.113596\pi$
$444$	0	0
$445$	1.59720e6i	0.382347i
$446$	0	0
$447$	−2.63493e6	−0.623735
$448$	0	0
$449$	−4.01250e6	−0.939288	−0.469644	−	0.882856i	$-0.655618\pi$
$449$	−4.01250e6	−0.939288	−0.469644	+	0.882856i	$0.655618\pi$
$450$	0	0
$451$	− 453210.i	− 0.104920i
$452$	0	0
$453$	1.22953e7i	2.81509i
$454$	0	0
$455$	−3.87284e6	−0.877002
$456$	0	0
$457$	−6.10953e6	−1.36841	−0.684206	−	0.729288i	$-0.739851\pi$
$457$	−6.10953e6	−1.36841	−0.684206	+	0.729288i	$0.739851\pi$
$458$	0	0
$459$	1.56911e6i	0.347634i
$460$	0	0
$461$	− 6.59208e6i	− 1.44467i	−0.691541	−	0.722337i	$-0.743068\pi$
$461$	− 6.59208e6i	− 1.44467i	0.691541	−	0.722337i	$-0.256932\pi$
$462$	0	0
$463$	2.37278e6	0.514406	0.257203	−	0.966357i	$-0.417199\pi$
$463$	2.37278e6	0.514406	0.257203	+	0.966357i	$0.417199\pi$
$464$	0	0
$465$	−3.48172e6	−0.746726
$466$	0	0
$467$	− 6.30012e6i	− 1.33677i	−0.743816	−	0.668385i	$-0.766986\pi$
$467$	− 6.30012e6i	− 1.33677i	0.743816	−	0.668385i	$-0.233014\pi$
$468$	0	0
$469$	− 1.71443e6i	− 0.359905i
$470$	0	0
$471$	1.47369e7	3.06094
$472$	0	0
$473$	975641.	0.200510
$474$	0	0
$475$	− 1.74432e6i	− 0.354726i
$476$	0	0
$477$	− 2.55390e6i	− 0.513934i
$478$	0	0
$479$	2.28649e6	0.455334	0.227667	−	0.973739i	$-0.426890\pi$
$479$	2.28649e6	0.455334	0.227667	+	0.973739i	$0.426890\pi$
$480$	0	0
$481$	3.45590e6	0.681081
$482$	0	0
$483$	1.97971e7i	3.86129i
$484$	0	0
$485$	359712.i	0.0694385i
$486$	0	0
$487$	−2.28593e6	−0.436758	−0.218379	−	0.975864i	$-0.570077\pi$
$487$	−2.28593e6	−0.436758	−0.218379	+	0.975864i	$0.570077\pi$
$488$	0	0
$489$	1.40458e7	2.65629
$490$	0	0
$491$	7.08998e6i	1.32721i	0.748081	+	0.663607i	$0.230975\pi$
$491$	7.08998e6i	1.32721i	−0.748081	+	0.663607i	$0.769025\pi$
$492$	0	0
$493$	− 1.63902e6i	− 0.303716i
$494$	0	0
$495$	832799.	0.152766
$496$	0	0
$497$	−1.32440e7	−2.40506
$498$	0	0
$499$	− 3.60881e6i	− 0.648804i	−0.945919	−	0.324402i	$-0.894837\pi$
$499$	− 3.60881e6i	− 0.648804i	0.945919	−	0.324402i	$-0.105163\pi$
$500$	0	0
$501$	1.81534e7i	3.23121i
$502$	0	0
$503$	1.01067e7	1.78111	0.890554	−	0.454878i	$-0.150317\pi$
$503$	1.01067e7	1.78111	0.890554	+	0.454878i	$0.150317\pi$
$504$	0	0
$505$	1.84404e6	0.321767
$506$	0	0
$507$	− 3.38919e6i	− 0.585566i
$508$	0	0
$509$	− 2.49518e6i	− 0.426882i	−0.976956	−	0.213441i	$-0.931533\pi$
$509$	− 2.49518e6i	− 0.426882i	0.976956	−	0.213441i	$-0.0684670\pi$
$510$	0	0
$511$	8.02045e6	1.35877
$512$	0	0
$513$	2.10974e7	3.53945
$514$	0	0
$515$	− 334363.i	− 0.0555520i
$516$	0	0
$517$	− 599380.i	− 0.0986225i
$518$	0	0
$519$	−6.42422e6	−1.04689
$520$	0	0
$521$	−9.37259e6	−1.51274	−0.756371	−	0.654142i	$-0.773030\pi$
$521$	−9.37259e6	−1.51274	−0.756371	+	0.654142i	$0.773030\pi$
$522$	0	0
$523$	− 1.96213e6i	− 0.313670i	−0.987625	−	0.156835i	$-0.949871\pi$
$523$	− 1.96213e6i	− 0.313670i	0.987625	−	0.156835i	$-0.0501291\pi$
$524$	0	0
$525$	− 3.79723e6i	− 0.601268i
$526$	0	0
$527$	1.04850e6	0.164453
$528$	0	0
$529$	4.18131e6	0.649640
$530$	0	0
$531$	− 1.98809e7i	− 3.05985i
$532$	0	0
$533$	− 4.94646e6i	− 0.754183i
$534$	0	0
$535$	5.15145e6	0.778117
$536$	0	0
$537$	−8.29674e6	−1.24157
$538$	0	0
$539$	− 2.04512e6i	− 0.303213i
$540$	0	0
$541$	2.57325e6i	0.377998i	0.981977	+	0.188999i	$0.0605243\pi$
$541$	2.57325e6i	0.377998i	−0.981977	+	0.188999i	$0.939476\pi$
$542$	0	0
$543$	1.49228e6	0.217195
$544$	0	0
$545$	1.82715e6	0.263502
$546$	0	0
$547$	− 6.99737e6i	− 0.999923i	−0.866048	−	0.499962i	$-0.833347\pi$
$547$	− 6.99737e6i	− 0.999923i	0.866048	−	0.499962i	$-0.166653\pi$
$548$	0	0
$549$	1.43670e7i	2.03440i
$550$	0	0
$551$	−2.20374e7	−3.09230
$552$	0	0
$553$	−8.10230e6	−1.12667
$554$	0	0
$555$	3.38843e6i	0.466946i
$556$	0	0
$557$	1.11329e7i	1.52044i	0.649665	+	0.760221i	$0.274909\pi$
$557$	1.11329e7i	1.52044i	−0.649665	+	0.760221i	$0.725091\pi$
$558$	0	0
$559$	1.06484e7	1.44130
$560$	0	0
$561$	−368630.	−0.0494520
$562$	0	0
$563$	7.19654e6i	0.956870i	0.878123	+	0.478435i	$0.158796\pi$
$563$	7.19654e6i	0.956870i	−0.878123	+	0.478435i	$0.841204\pi$
$564$	0	0
$565$	3.87273e6i	0.510383i
$566$	0	0
$567$	1.82339e7	2.38190
$568$	0	0
$569$	3.80510e6	0.492703	0.246352	−	0.969181i	$-0.420768\pi$
$569$	3.80510e6	0.492703	0.246352	+	0.969181i	$0.420768\pi$
$570$	0	0
$571$	− 3.56459e6i	− 0.457530i	−0.973482	−	0.228765i	$-0.926531\pi$
$571$	− 3.56459e6i	− 0.457530i	0.973482	−	0.228765i	$-0.0734687\pi$
$572$	0	0
$573$	− 1.15102e7i	− 1.46452i
$574$	0	0
$575$	−2.03655e6	−0.256877
$576$	0	0
$577$	1.33424e7	1.66837	0.834187	−	0.551482i	$-0.185937\pi$
$577$	1.33424e7	1.66837	0.834187	+	0.551482i	$0.185937\pi$
$578$	0	0
$579$	1.18816e7i	1.47292i
$580$	0	0
$581$	− 4.65682e6i	− 0.572333i
$582$	0	0
$583$	318075.	0.0387578
$584$	0	0
$585$	9.08941e6	1.09811
$586$	0	0
$587$	1.02312e7i	1.22555i	0.790257	+	0.612775i	$0.209947\pi$
$587$	1.02312e7i	1.22555i	−0.790257	+	0.612775i	$0.790053\pi$
$588$	0	0
$589$	− 1.40976e7i	− 1.67439i
$590$	0	0
$591$	−7.97545e6	−0.939261
$592$	0	0
$593$	1.50572e7	1.75836	0.879181	−	0.476487i	$-0.158090\pi$
$593$	1.50572e7	1.75836	0.879181	+	0.476487i	$0.158090\pi$
$594$	0	0
$595$	1.14351e6i	0.132418i
$596$	0	0
$597$	− 2.23988e7i	− 2.57211i
$598$	0	0
$599$	994780.	0.113282	0.0566409	−	0.998395i	$-0.481961\pi$
$599$	994780.	0.113282	0.0566409	+	0.998395i	$0.481961\pi$
$600$	0	0
$601$	6.59335e6	0.744595	0.372297	−	0.928113i	$-0.378570\pi$
$601$	6.59335e6	0.744595	0.372297	+	0.928113i	$0.378570\pi$
$602$	0	0
$603$	4.02371e6i	0.450644i
$604$	0	0
$605$	− 3.92255e6i	− 0.435693i
$606$	0	0
$607$	−9.09559e6	−1.00198	−0.500990	−	0.865453i	$-0.667031\pi$
$607$	−9.09559e6	−1.00198	−0.500990	+	0.865453i	$0.667031\pi$
$608$	0	0
$609$	−4.79734e7	−5.24152
$610$	0	0
$611$	− 6.54181e6i	− 0.708916i
$612$	0	0
$613$	− 1.08785e7i	− 1.16927i	−0.811295	−	0.584637i	$-0.801237\pi$
$613$	− 1.08785e7i	− 1.16927i	0.811295	−	0.584637i	$-0.198763\pi$
$614$	0	0
$615$	4.84989e6	0.517064
$616$	0	0
$617$	−2.35827e6	−0.249391	−0.124696	−	0.992195i	$-0.539795\pi$
$617$	−2.35827e6	−0.249391	−0.124696	+	0.992195i	$0.539795\pi$
$618$	0	0
$619$	2.95345e6i	0.309815i	0.987929	+	0.154907i	$0.0495079\pi$
$619$	2.95345e6i	0.309815i	−0.987929	+	0.154907i	$0.950492\pi$
$620$	0	0
$621$	− 2.46318e7i	− 2.56311i
$622$	0	0
$623$	−1.40782e7	−1.45321
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	4.95640e6i	0.503498i
$628$	0	0
$629$	− 1.02041e6i	− 0.102836i
$630$	0	0
$631$	1.49811e7	1.49785	0.748927	−	0.662653i	$-0.230570\pi$
$631$	1.49811e7	1.49785	0.748927	+	0.662653i	$0.230570\pi$
$632$	0	0
$633$	−6.28481e6	−0.623423
$634$	0	0
$635$	− 3.91950e6i	− 0.385742i
$636$	0	0
$637$	− 2.23211e7i	− 2.17955i
$638$	0	0
$639$	3.10831e7	3.01143
$640$	0	0
$641$	−4.70605e6	−0.452388	−0.226194	−	0.974082i	$-0.572628\pi$
$641$	−4.70605e6	−0.452388	−0.226194	+	0.974082i	$0.572628\pi$
$642$	0	0
$643$	1.00759e7i	0.961074i	0.876974	+	0.480537i	$0.159558\pi$
$643$	1.00759e7i	0.961074i	−0.876974	+	0.480537i	$0.840442\pi$
$644$	0	0
$645$	1.04405e7i	0.988151i
$646$	0	0
$647$	1.69585e6	0.159267	0.0796337	−	0.996824i	$-0.474625\pi$
$647$	1.69585e6	0.159267	0.0796337	+	0.996824i	$0.474625\pi$
$648$	0	0
$649$	2.47607e6	0.230755
$650$	0	0
$651$	− 3.06891e7i	− 2.83812i
$652$	0	0
$653$	− 9.63786e6i	− 0.884500i	−0.896892	−	0.442250i	$-0.854180\pi$
$653$	− 9.63786e6i	− 0.884500i	0.896892	−	0.442250i	$-0.145820\pi$
$654$	0	0
$655$	−6.03993e6	−0.550083
$656$	0	0
$657$	−1.88237e7	−1.70134
$658$	0	0
$659$	− 9.50539e6i	− 0.852622i	−0.904577	−	0.426311i	$-0.859813\pi$
$659$	− 9.50539e6i	− 0.852622i	0.904577	−	0.426311i	$-0.140187\pi$
$660$	0	0
$661$	− 1.34346e7i	− 1.19597i	−0.801507	−	0.597986i	$-0.795968\pi$
$661$	− 1.34346e7i	− 1.19597i	0.801507	−	0.597986i	$-0.204032\pi$
$662$	0	0
$663$	−4.02334e6	−0.355470
$664$	0	0
$665$	1.53750e7	1.34822
$666$	0	0
$667$	2.57293e7i	2.23931i
$668$	0	0
$669$	1.02390e7i	0.884493i
$670$	0	0
$671$	−1.78935e6	−0.153422
$672$	0	0
$673$	930663.	0.0792054	0.0396027	−	0.999216i	$-0.487391\pi$
$673$	930663.	0.0792054	0.0396027	+	0.999216i	$0.487391\pi$
$674$	0	0
$675$	4.72458e6i	0.399120i
$676$	0	0
$677$	6.87737e6i	0.576701i	0.957525	+	0.288350i	$0.0931068\pi$
$677$	6.87737e6i	0.576701i	−0.957525	+	0.288350i	$0.906893\pi$
$678$	0	0
$679$	−3.17062e6	−0.263919
$680$	0	0
$681$	2.09997e7	1.73518
$682$	0	0
$683$	909803.i	0.0746269i	0.999304	+	0.0373135i	$0.0118800\pi$
$683$	909803.i	0.0746269i	−0.999304	+	0.0373135i	$0.988120\pi$
$684$	0	0
$685$	867960.i	0.0706762i
$686$	0	0
$687$	−8.97169e6	−0.725241
$688$	0	0
$689$	3.47157e6	0.278598
$690$	0	0
$691$	− 2.04678e7i	− 1.63071i	−0.578961	−	0.815355i	$-0.696542\pi$
$691$	− 2.04678e7i	− 1.63071i	0.578961	−	0.815355i	$-0.303458\pi$
$692$	0	0
$693$	7.34058e6i	0.580627i
$694$	0	0
$695$	−3.71304e6	−0.291587
$696$	0	0
$697$	−1.46052e6	−0.113874
$698$	0	0
$699$	− 1.45490e7i	− 1.12627i
$700$	0	0
$701$	− 1.75532e7i	− 1.34916i	−0.738203	−	0.674578i	$-0.764326\pi$
$701$	− 1.75532e7i	− 1.34916i	0.738203	−	0.674578i	$-0.235674\pi$
$702$	0	0
$703$	−1.37198e7	−1.04703
$704$	0	0
$705$	6.41410e6	0.486030
$706$	0	0
$707$	1.62540e7i	1.22296i
$708$	0	0
$709$	− 2.23525e7i	− 1.66997i	−0.550270	−	0.834987i	$-0.685475\pi$
$709$	− 2.23525e7i	− 1.66997i	0.550270	−	0.834987i	$-0.314525\pi$
$710$	0	0
$711$	1.90158e7	1.41072
$712$	0	0
$713$	−1.64593e7	−1.21252
$714$	0	0
$715$	1.13204e6i	0.0828128i
$716$	0	0
$717$	3.01154e7i	2.18771i
$718$	0	0
$719$	−1.84982e7	−1.33447	−0.667234	−	0.744848i	$-0.732522\pi$
$719$	−1.84982e7	−1.33447	−0.667234	+	0.744848i	$0.732522\pi$
$720$	0	0
$721$	2.94719e6	0.211140
$722$	0	0
$723$	− 1.74709e6i	− 0.124300i
$724$	0	0
$725$	− 4.93508e6i	− 0.348698i
$726$	0	0
$727$	5.77823e6	0.405470	0.202735	−	0.979234i	$-0.435017\pi$
$727$	5.77823e6	0.405470	0.202735	+	0.979234i	$0.435017\pi$
$728$	0	0
$729$	7.85162e6	0.547193
$730$	0	0
$731$	− 3.14410e6i	− 0.217622i
$732$	0	0
$733$	− 1.00639e6i	− 0.0691839i	−0.999402	−	0.0345920i	$-0.988987\pi$
$733$	− 1.00639e6i	− 0.0691839i	0.999402	−	0.0345920i	$-0.0110132\pi$
$734$	0	0
$735$	2.18853e7	1.49429
$736$	0	0
$737$	−501133.	−0.0339848
$738$	0	0
$739$	− 1.10508e7i	− 0.744357i	−0.928161	−	0.372179i	$-0.878611\pi$
$739$	− 1.10508e7i	− 0.744357i	0.928161	−	0.372179i	$-0.121389\pi$
$740$	0	0
$741$	5.40956e7i	3.61923i
$742$	0	0
$743$	6.91879e6	0.459789	0.229894	−	0.973216i	$-0.426162\pi$
$743$	6.91879e6	0.459789	0.229894	+	0.973216i	$0.426162\pi$
$744$	0	0
$745$	2.38920e6	0.157711
$746$	0	0
$747$	1.09294e7i	0.716630i
$748$	0	0
$749$	4.54066e7i	2.95743i
$750$	0	0
$751$	1.79833e7	1.16351	0.581756	−	0.813364i	$-0.302366\pi$
$751$	1.79833e7	1.16351	0.581756	+	0.813364i	$0.302366\pi$
$752$	0	0
$753$	4.40642e7	2.83203
$754$	0	0
$755$	− 1.11486e7i	− 0.711795i
$756$	0	0
$757$	− 2.59762e7i	− 1.64754i	−0.566923	−	0.823771i	$-0.691866\pi$
$757$	− 2.59762e7i	− 1.64754i	0.566923	−	0.823771i	$-0.308134\pi$
$758$	0	0
$759$	5.78674e6	0.364611
$760$	0	0
$761$	8.18122e6	0.512102	0.256051	−	0.966663i	$-0.417579\pi$
$761$	8.18122e6	0.512102	0.256051	+	0.966663i	$0.417579\pi$
$762$	0	0
$763$	1.61052e7i	1.00151i
$764$	0	0
$765$	− 2.68378e6i	− 0.165804i
$766$	0	0
$767$	2.70246e7	1.65871
$768$	0	0
$769$	−1.85187e7	−1.12926	−0.564630	−	0.825344i	$-0.690981\pi$
$769$	−1.85187e7	−1.12926	−0.564630	+	0.825344i	$0.690981\pi$
$770$	0	0
$771$	− 1.31889e7i	− 0.799049i
$772$	0	0
$773$	− 1.07043e7i	− 0.644329i	−0.946684	−	0.322164i	$-0.895590\pi$
$773$	− 1.07043e7i	− 0.644329i	0.946684	−	0.322164i	$-0.104410\pi$
$774$	0	0
$775$	3.15702e6	0.188809
$776$	0	0
$777$	−2.98668e7	−1.77475
$778$	0	0
$779$	1.96373e7i	1.15941i
$780$	0	0
$781$	3.87125e6i	0.227104i
$782$	0	0
$783$	5.96894e7	3.47931
$784$	0	0
$785$	−1.33626e7	−0.773955
$786$	0	0
$787$	68571.9i	0.00394648i	0.999998	+	0.00197324i	$0.000628102\pi$
$787$	68571.9i	0.00394648i	−0.999998	+	0.00197324i	$0.999372\pi$
$788$	0	0
$789$	− 1.98523e7i	− 1.13532i
$790$	0	0
$791$	−3.41356e7	−1.93984
$792$	0	0
$793$	−1.95294e7	−1.10283
$794$	0	0
$795$	3.40379e6i	0.191005i
$796$	0	0
$797$	2.51343e7i	1.40159i	0.713363	+	0.700794i	$0.247171\pi$
$797$	2.51343e7i	1.40159i	−0.713363	+	0.700794i	$0.752829\pi$
$798$	0	0
$799$	−1.93157e6	−0.107039
$800$	0	0
$801$	3.30411e7	1.81959
$802$	0	0
$803$	− 2.34440e6i	− 0.128305i
$804$	0	0
$805$	− 1.79508e7i	− 0.976325i
$806$	0	0
$807$	−4.81946e7	−2.60504
$808$	0	0
$809$	2.45606e7	1.31937	0.659687	−	0.751540i	$-0.270689\pi$
$809$	2.45606e7	1.31937	0.659687	+	0.751540i	$0.270689\pi$
$810$	0	0
$811$	− 1.94878e6i	− 0.104043i	−0.998646	−	0.0520213i	$-0.983434\pi$
$811$	− 1.94878e6i	− 0.104043i	0.998646	−	0.0520213i	$-0.0165664\pi$
$812$	0	0
$813$	4.97123e7i	2.63778i
$814$	0	0
$815$	−1.27360e7	−0.671641
$816$	0	0
$817$	−4.22739e7	−2.21573
$818$	0	0
$819$	8.01172e7i	4.17365i
$820$	0	0
$821$	− 2.02265e7i	− 1.04728i	−0.851940	−	0.523639i	$-0.824574\pi$
$821$	− 2.02265e7i	− 1.04728i	0.851940	−	0.523639i	$-0.175426\pi$
$822$	0	0
$823$	1.57978e7	0.813010	0.406505	−	0.913649i	$-0.366747\pi$
$823$	1.57978e7	0.813010	0.406505	+	0.913649i	$0.366747\pi$
$824$	0	0
$825$	−1.10994e6	−0.0567761
$826$	0	0
$827$	− 2.99880e7i	− 1.52470i	−0.647167	−	0.762348i	$-0.724046\pi$
$827$	− 2.99880e7i	− 1.52470i	0.647167	−	0.762348i	$-0.275954\pi$
$828$	0	0
$829$	− 1.30834e7i	− 0.661201i	−0.943771	−	0.330601i	$-0.892749\pi$
$829$	− 1.30834e7i	− 0.661201i	0.943771	−	0.330601i	$-0.107251\pi$
$830$	0	0
$831$	−6.70290e6	−0.336713
$832$	0	0
$833$	−6.59063e6	−0.329090
$834$	0	0
$835$	− 1.64605e7i	− 0.817008i
$836$	0	0
$837$	3.81839e7i	1.88394i
$838$	0	0
$839$	−1.58503e7	−0.777378	−0.388689	−	0.921369i	$-0.627072\pi$
$839$	−1.58503e7	−0.777378	−0.388689	+	0.921369i	$0.627072\pi$
$840$	0	0
$841$	−4.18377e7	−2.03975
$842$	0	0
$843$	− 1.04039e7i	− 0.504227i
$844$	0	0
$845$	3.07312e6i	0.148060i
$846$	0	0
$847$	3.45747e7	1.65596
$848$	0	0
$849$	−2.49440e7	−1.18767
$850$	0	0
$851$	1.60183e7i	0.758215i
$852$	0	0
$853$	8.25379e6i	0.388402i	0.980962	+	0.194201i	$0.0622113\pi$
$853$	8.25379e6i	0.388402i	−0.980962	+	0.194201i	$0.937789\pi$
$854$	0	0
$855$	−3.60847e7	−1.68814
$856$	0	0
$857$	1.20306e7	0.559546	0.279773	−	0.960066i	$-0.409741\pi$
$857$	1.20306e7	0.559546	0.279773	+	0.960066i	$0.409741\pi$
$858$	0	0
$859$	− 7.28563e6i	− 0.336887i	−0.985711	−	0.168443i	$-0.946126\pi$
$859$	− 7.28563e6i	− 0.336887i	0.985711	−	0.168443i	$-0.0538741\pi$
$860$	0	0
$861$	4.27486e7i	1.96523i
$862$	0	0
$863$	−2.70168e7	−1.23483	−0.617415	−	0.786638i	$-0.711820\pi$
$863$	−2.70168e7	−1.23483	−0.617415	+	0.786638i	$0.711820\pi$
$864$	0	0
$865$	5.82511e6	0.264706
$866$	0	0
$867$	− 3.79593e7i	− 1.71502i
$868$	0	0
$869$	2.36833e6i	0.106388i
$870$	0	0
$871$	−5.46952e6	−0.244289
$872$	0	0
$873$	7.44134e6	0.330458
$874$	0	0
$875$	3.44310e6i	0.152030i
$876$	0	0
$877$	− 1.20510e7i	− 0.529082i	−0.964374	−	0.264541i	$-0.914780\pi$
$877$	− 1.20510e7i	− 0.529082i	0.964374	−	0.264541i	$-0.0852204\pi$
$878$	0	0
$879$	−4.33569e7	−1.89272
$880$	0	0
$881$	4.10240e7	1.78073	0.890366	−	0.455245i	$-0.150448\pi$
$881$	4.10240e7	1.78073	0.890366	+	0.455245i	$0.150448\pi$
$882$	0	0
$883$	− 2.77485e6i	− 0.119767i	−0.998205	−	0.0598835i	$-0.980927\pi$
$883$	− 2.77485e6i	− 0.119767i	0.998205	−	0.0598835i	$-0.0190729\pi$
$884$	0	0
$885$	2.64970e7i	1.13720i
$886$	0	0
$887$	−2.74930e7	−1.17331	−0.586655	−	0.809837i	$-0.699556\pi$
$887$	−2.74930e7	−1.17331	−0.586655	+	0.809837i	$0.699556\pi$
$888$	0	0
$889$	3.45479e7	1.46611
$890$	0	0
$891$	− 5.32984e6i	− 0.224916i
$892$	0	0
$893$	2.59708e7i	1.08982i
$894$	0	0
$895$	7.52300e6	0.313931
$896$	0	0
$897$	6.31582e7	2.62089
$898$	0	0
$899$	− 3.98852e7i	− 1.64593i
$900$	0	0
$901$	− 1.02503e6i	− 0.0420654i
$902$	0	0
$903$	−9.20264e7	−3.75572
$904$	0	0
$905$	−1.35311e6	−0.0549177
$906$	0	0
$907$	− 1.59931e7i	− 0.645525i	−0.946480	−	0.322763i	$-0.895388\pi$
$907$	− 1.59931e7i	− 0.645525i	0.946480	−	0.322763i	$-0.104612\pi$
$908$	0	0
$909$	− 3.81476e7i	− 1.53129i
$910$	0	0
$911$	6.65270e6	0.265584	0.132792	−	0.991144i	$-0.457606\pi$
$911$	6.65270e6	0.265584	0.132792	+	0.991144i	$0.457606\pi$
$912$	0	0
$913$	−1.36120e6	−0.0540438
$914$	0	0
$915$	− 1.91482e7i	− 0.756092i
$916$	0	0
$917$	− 5.32380e7i	− 2.09073i
$918$	0	0
$919$	−2.78769e7	−1.08882	−0.544409	−	0.838820i	$-0.683246\pi$
$919$	−2.78769e7	−1.08882	−0.544409	+	0.838820i	$0.683246\pi$
$920$	0	0
$921$	−4.14931e7	−1.61186
$922$	0	0
$923$	4.22520e7i	1.63246i
$924$	0	0
$925$	− 3.07243e6i	− 0.118067i
$926$	0	0
$927$	−6.91695e6	−0.264372
$928$	0	0
$929$	2.07437e7	0.788581	0.394290	−	0.918986i	$-0.370990\pi$
$929$	2.07437e7	0.788581	0.394290	+	0.918986i	$0.370990\pi$
$930$	0	0
$931$	8.86140e7i	3.35064i
$932$	0	0
$933$	1.40381e6i	0.0527963i
$934$	0	0
$935$	334252.	0.0125039
$936$	0	0
$937$	−1.13976e7	−0.424097	−0.212049	−	0.977259i	$-0.568014\pi$
$937$	−1.13976e7	−0.424097	−0.212049	+	0.977259i	$0.568014\pi$
$938$	0	0
$939$	− 2.89872e7i	− 1.07286i
$940$	0	0
$941$	− 995880.i	− 0.0366634i	−0.999832	−	0.0183317i	$-0.994165\pi$
$941$	− 995880.i	− 0.0366634i	0.999832	−	0.0183317i	$-0.00583549\pi$
$942$	0	0
$943$	2.29271e7	0.839596
$944$	0	0
$945$	−4.16440e7	−1.51696
$946$	0	0
$947$	− 5.26122e6i	− 0.190639i	−0.995447	−	0.0953194i	$-0.969613\pi$
$947$	− 5.26122e6i	− 0.190639i	0.995447	−	0.0953194i	$-0.0303872\pi$
$948$	0	0
$949$	− 2.55875e7i	− 0.922280i
$950$	0	0
$951$	−1.62948e7	−0.584247
$952$	0	0
$953$	−2.85407e7	−1.01797	−0.508983	−	0.860777i	$-0.669978\pi$
$953$	−2.85407e7	−1.01797	−0.508983	+	0.860777i	$0.669978\pi$
$954$	0	0
$955$	1.04367e7i	0.370302i
$956$	0	0
$957$	1.40228e7i	0.494942i
$958$	0	0
$959$	−7.65050e6	−0.268623
$960$	0	0
$961$	−3.11422e6	−0.108778
$962$	0	0
$963$	− 1.06568e8i	− 3.70306i
$964$	0	0
$965$	− 1.07735e7i	− 0.372426i
$966$	0	0
$967$	8.62711e6	0.296688	0.148344	−	0.988936i	$-0.452606\pi$
$967$	8.62711e6	0.296688	0.148344	+	0.988936i	$0.452606\pi$
$968$	0	0
$969$	1.59725e7	0.546467
$970$	0	0
$971$	− 1.69141e7i	− 0.575706i	−0.957675	−	0.287853i	$-0.907058\pi$
$971$	− 1.69141e7i	− 0.575706i	0.957675	−	0.287853i	$-0.0929415\pi$
$972$	0	0
$973$	− 3.27280e7i	− 1.10825i
$974$	0	0
$975$	−1.21142e7	−0.408117
$976$	0	0
$977$	−1.09583e7	−0.367288	−0.183644	−	0.982993i	$-0.558789\pi$
$977$	−1.09583e7	−0.367288	−0.183644	+	0.982993i	$0.558789\pi$
$978$	0	0
$979$	4.11511e6i	0.137222i
$980$	0	0
$981$	− 3.77982e7i	− 1.25400i
$982$	0	0
$983$	1.08936e7	0.359573	0.179787	−	0.983706i	$-0.442459\pi$
$983$	1.08936e7	0.359573	0.179787	+	0.983706i	$0.442459\pi$
$984$	0	0
$985$	7.23167e6	0.237491
$986$	0	0
$987$	5.65360e7i	1.84728i
$988$	0	0
$989$	4.93561e7i	1.60454i
$990$	0	0
$991$	1.60015e6	0.0517578	0.0258789	−	0.999665i	$-0.491762\pi$
$991$	1.60015e6	0.0517578	0.0258789	+	0.999665i	$0.491762\pi$
$992$	0	0
$993$	−4.07367e7	−1.31103
$994$	0	0
$995$	2.03099e7i	0.650355i
$996$	0	0
$997$	− 1.42051e7i	− 0.452593i	−0.974059	−	0.226296i	$-0.927338\pi$
$997$	− 1.42051e7i	− 0.452593i	0.974059	−	0.226296i	$-0.0726618\pi$
$998$	0	0
$999$	3.71608e7	1.17807

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.6.d.b.161.8	yes	8
4.3	odd	2		320.6.d.a.161.1	✓	8
8.3	odd	2		320.6.d.a.161.8	yes	8
8.5	even	2	inner	320.6.d.b.161.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
320.6.d.a.161.1	✓	8	4.3	odd	2
320.6.d.a.161.8	yes	8	8.3	odd	2
320.6.d.b.161.1	yes	8	8.5	even	2	inner
320.6.d.b.161.8	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+27.5713i q^{3} -25.0000i q^{5} +220.359 q^{7} -517.174 q^{9} +O(q^{10})\)	\(q+27.5713i q^{3} -25.0000i q^{5} +220.359 q^{7} -517.174 q^{9} -64.4115i q^{11} -703.006i q^{13} +689.281 q^{15} -207.573 q^{17} +2790.91i q^{19} +6075.56i q^{21} +3258.47 q^{23} -625.000 q^{25} -7559.32i q^{27} +7896.13i q^{29} -5051.23 q^{31} +1775.91 q^{33} -5508.96i q^{35} +4915.89i q^{37} +19382.8 q^{39} +7036.16 q^{41} +15147.0i q^{43} +12929.4i q^{45} +9305.48 q^{47} +31750.9 q^{49} -5723.05i q^{51} +4938.18i q^{53} -1610.29 q^{55} -76949.0 q^{57} +38441.5i q^{59} -27779.9i q^{61} -113964. q^{63} -17575.2 q^{65} -7780.18i q^{67} +89840.2i q^{69} -60101.8 q^{71} +36397.3 q^{73} -17232.0i q^{75} -14193.6i q^{77} -36768.7 q^{79} +82746.7 q^{81} -21132.9i q^{83} +5189.32i q^{85} -217706. q^{87} -63887.8 q^{89} -154913. i q^{91} -139269. i q^{93} +69772.9 q^{95} -14388.5 q^{97} +33312.0i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 320 q^{7} - 1192 q^{9}+O(q^{10}) \) 8 * q + 320 * q^7 - 1192 * q^9	\( 8 q + 320 q^{7} - 1192 q^{9} - 2400 q^{17} - 1760 q^{23} - 5000 q^{25} - 31040 q^{31} + 3760 q^{33} - 4480 q^{39} + 21584 q^{41} - 47680 q^{47} + 82824 q^{49} - 18400 q^{55} - 106640 q^{57} - 322400 q^{63} - 44000 q^{65} - 246720 q^{71} + 46400 q^{73} - 325760 q^{79} - 82328 q^{81} - 636320 q^{87} - 78192 q^{89} + 40800 q^{95} + 24960 q^{97}+O(q^{100}) \) 8 * q + 320 * q^7 - 1192 * q^9 - 2400 * q^17 - 1760 * q^23 - 5000 * q^25 - 31040 * q^31 + 3760 * q^33 - 4480 * q^39 + 21584 * q^41 - 47680 * q^47 + 82824 * q^49 - 18400 * q^55 - 106640 * q^57 - 322400 * q^63 - 44000 * q^65 - 246720 * q^71 + 46400 * q^73 - 325760 * q^79 - 82328 * q^81 - 636320 * q^87 - 78192 * q^89 + 40800 * q^95 + 24960 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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