LMFDB - Embedded newform 441.6.a.bb.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,6,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("441.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	441.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:	$70.7292645375$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 $ x^6 - 2x^5 - 59x^4 + 122x^3 + 941x^2 - 1856*x - 2338
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{3} $
Twist minimal:	no (minimal twist has level 147)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.910122$ of defining polynomial
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.31815 q^{2} -3.71724 q^{4} +103.471 q^{5} -189.950 q^{8} +O(q^{10})$	$q+5.31815 q^{2} -3.71724 q^{4} +103.471 q^{5} -189.950 q^{8} +550.272 q^{10} -653.308 q^{11} +138.055 q^{13} -891.231 q^{16} +1174.38 q^{17} +1710.77 q^{19} -384.625 q^{20} -3474.39 q^{22} +4020.28 q^{23} +7581.15 q^{25} +734.196 q^{26} -2649.46 q^{29} +2874.65 q^{31} +1338.69 q^{32} +6245.51 q^{34} +2856.87 q^{37} +9098.15 q^{38} -19654.2 q^{40} +216.487 q^{41} +2928.29 q^{43} +2428.50 q^{44} +21380.5 q^{46} +14816.8 q^{47} +40317.7 q^{50} -513.182 q^{52} -21167.7 q^{53} -67598.1 q^{55} -14090.3 q^{58} +34689.1 q^{59} +8753.12 q^{61} +15287.8 q^{62} +35638.7 q^{64} +14284.6 q^{65} -12068.5 q^{67} -4365.43 q^{68} +35541.5 q^{71} -33485.8 q^{73} +15193.3 q^{74} -6359.35 q^{76} +43134.8 q^{79} -92216.1 q^{80} +1151.31 q^{82} -43338.8 q^{83} +121513. q^{85} +15573.1 q^{86} +124096. q^{88} -103533. q^{89} -14944.3 q^{92} +78798.1 q^{94} +177015. q^{95} +86294.4 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) $ 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8	$ 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97}+O(q^{100}) $ 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8 - 864 * q^10 - 604 * q^11 - 1352 * q^13 + 4578 * q^16 + 3028 * q^17 - 1728 * q^19 + 452 * q^20 - 4116 * q^22 + 4484 * q^23 + 4806 * q^25 + 14172 * q^26 + 5320 * q^29 - 3976 * q^31 + 37326 * q^32 + 16336 * q^34 + 22680 * q^37 + 52744 * q^38 - 100600 * q^40 + 28756 * q^41 - 6768 * q^43 + 64940 * q^44 + 540 * q^46 + 51552 * q^47 + 40622 * q^50 - 119296 * q^52 - 80884 * q^53 - 11656 * q^55 - 70464 * q^58 + 8872 * q^59 - 50896 * q^61 + 11824 * q^62 + 199590 * q^64 - 3492 * q^65 + 6480 * q^67 + 37348 * q^68 + 110852 * q^71 - 64232 * q^73 + 27464 * q^74 + 194864 * q^76 + 111696 * q^79 - 308940 * q^80 + 189640 * q^82 + 101128 * q^83 - 23292 * q^85 - 3824 * q^86 - 97788 * q^88 - 35012 * q^89 + 449260 * q^92 + 121016 * q^94 + 119080 * q^95 - 70952 * q^97

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 $

$2$	5.31815	0.940126	0.470063	−	0.882633i	$-0.344231\pi$
$2$	5.31815	0.940126	0.470063	+	0.882633i	$0.344231\pi$
$3$	0	0
$3$	0	0
$4$	−3.71724	−0.116164
$5$	103.471	1.85094	0.925468	−	0.378825i	$-0.123672\pi$
$5$	103.471	1.85094	0.925468	+	0.378825i	$0.123672\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−189.950	−1.04933
$9$	0	0
$10$	550.272	1.74011
$11$	−653.308	−1.62793	−0.813966	−	0.580912i	$-0.802696\pi$
$11$	−653.308	−1.62793	−0.813966	+	0.580912i	$0.802696\pi$
$12$	0	0
$13$	138.055	0.226565	0.113282	−	0.993563i	$-0.463863\pi$
$13$	138.055	0.226565	0.113282	+	0.993563i	$0.463863\pi$
$14$	0	0
$15$	0	0
$16$	−891.231	−0.870342
$17$	1174.38	0.985564	0.492782	−	0.870153i	$-0.335980\pi$
$17$	1174.38	0.985564	0.492782	+	0.870153i	$0.335980\pi$
$18$	0	0
$19$	1710.77	1.08720	0.543599	−	0.839345i	$-0.317061\pi$
$19$	1710.77	1.08720	0.543599	+	0.839345i	$0.317061\pi$
$20$	−384.625	−0.215012
$21$	0	0
$22$	−3474.39	−1.53046
$23$	4020.28	1.58466	0.792332	−	0.610091i	$-0.208867\pi$
$23$	4020.28	1.58466	0.792332	+	0.610091i	$0.208867\pi$
$24$	0	0
$25$	7581.15	2.42597
$26$	734.196	0.213000
$27$	0	0
$28$	0	0
$29$	−2649.46	−0.585009	−0.292505	−	0.956264i	$-0.594489\pi$
$29$	−2649.46	−0.585009	−0.292505	+	0.956264i	$0.594489\pi$
$30$	0	0
$31$	2874.65	0.537255	0.268628	−	0.963244i	$-0.413430\pi$
$31$	2874.65	0.537255	0.268628	+	0.963244i	$0.413430\pi$
$32$	1338.69	0.231103
$33$	0	0
$34$	6245.51	0.926554
$35$	0	0
$36$	0	0
$37$	2856.87	0.343073	0.171536	−	0.985178i	$-0.445127\pi$
$37$	2856.87	0.343073	0.171536	+	0.985178i	$0.445127\pi$
$38$	9098.15	1.02210
$39$	0	0
$40$	−19654.2	−1.94225
$41$	216.487	0.0201128	0.0100564	−	0.999949i	$-0.496799\pi$
$41$	216.487	0.0201128	0.0100564	+	0.999949i	$0.496799\pi$
$42$	0	0
$43$	2928.29	0.241514	0.120757	−	0.992682i	$-0.461468\pi$
$43$	2928.29	0.241514	0.120757	+	0.992682i	$0.461468\pi$
$44$	2428.50	0.189107
$45$	0	0
$46$	21380.5	1.48978
$47$	14816.8	0.978386	0.489193	−	0.872176i	$-0.337291\pi$
$47$	14816.8	0.978386	0.489193	+	0.872176i	$0.337291\pi$
$48$	0	0
$49$	0	0
$50$	40317.7	2.28071
$51$	0	0
$52$	−513.182	−0.0263186
$53$	−21167.7	−1.03510	−0.517552	−	0.855652i	$-0.673157\pi$
$53$	−21167.7	−1.03510	−0.517552	+	0.855652i	$0.673157\pi$
$54$	0	0
$55$	−67598.1	−3.01320
$56$	0	0
$57$	0	0
$58$	−14090.3	−0.549982
$59$	34689.1	1.29737	0.648684	−	0.761058i	$-0.275320\pi$
$59$	34689.1	1.29737	0.648684	+	0.761058i	$0.275320\pi$
$60$	0	0
$61$	8753.12	0.301189	0.150594	−	0.988596i	$-0.451881\pi$
$61$	8753.12	0.301189	0.150594	+	0.988596i	$0.451881\pi$
$62$	15287.8	0.505087
$63$	0	0
$64$	35638.7	1.08761
$65$	14284.6	0.419357
$66$	0	0
$67$	−12068.5	−0.328448	−0.164224	−	0.986423i	$-0.552512\pi$
$67$	−12068.5	−0.328448	−0.164224	+	0.986423i	$0.552512\pi$
$68$	−4365.43	−0.114487
$69$	0	0
$70$	0	0
$71$	35541.5	0.836738	0.418369	−	0.908277i	$-0.362602\pi$
$71$	35541.5	0.836738	0.418369	+	0.908277i	$0.362602\pi$
$72$	0	0
$73$	−33485.8	−0.735450	−0.367725	−	0.929935i	$-0.619863\pi$
$73$	−33485.8	−0.735450	−0.367725	+	0.929935i	$0.619863\pi$
$74$	15193.3	0.322531
$75$	0	0
$76$	−6359.35	−0.126293
$77$	0	0
$78$	0	0
$79$	43134.8	0.777607	0.388804	−	0.921321i	$-0.372888\pi$
$79$	43134.8	0.777607	0.388804	+	0.921321i	$0.372888\pi$
$80$	−92216.1	−1.61095
$81$	0	0
$82$	1151.31	0.0189086
$83$	−43338.8	−0.690529	−0.345265	−	0.938505i	$-0.612211\pi$
$83$	−43338.8	−0.690529	−0.345265	+	0.938505i	$0.612211\pi$
$84$	0	0
$85$	121513.	1.82422
$86$	15573.1	0.227054
$87$	0	0
$88$	124096.	1.70825
$89$	−103533.	−1.38550	−0.692748	−	0.721180i	$-0.743600\pi$
$89$	−103533.	−1.38550	−0.692748	+	0.721180i	$0.743600\pi$
$90$	0	0
$91$	0	0
$92$	−14944.3	−0.184080
$93$	0	0
$94$	78798.1	0.919806
$95$	177015.	2.01233
$96$	0	0
$97$	86294.4	0.931223	0.465611	−	0.884989i	$-0.345835\pi$
$97$	86294.4	0.931223	0.465611	+	0.884989i	$0.345835\pi$
$98$	0	0
$99$	0	0
$100$	−28180.9	−0.281809
$101$	182677.	1.78189	0.890944	−	0.454114i	$-0.150044\pi$
$101$	182677.	1.78189	0.890944	+	0.454114i	$0.150044\pi$
$102$	0	0
$103$	40088.2	0.372326	0.186163	−	0.982519i	$-0.440395\pi$
$103$	40088.2	0.372326	0.186163	+	0.982519i	$0.440395\pi$
$104$	−26223.5	−0.237742
$105$	0	0
$106$	−112573.	−0.973129
$107$	110207.	0.930575	0.465287	−	0.885160i	$-0.345951\pi$
$107$	110207.	0.930575	0.465287	+	0.885160i	$0.345951\pi$
$108$	0	0
$109$	−222354.	−1.79258	−0.896289	−	0.443471i	$-0.853747\pi$
$109$	−222354.	−1.79258	−0.896289	+	0.443471i	$0.853747\pi$
$110$	−359497.	−2.83279
$111$	0	0
$112$	0	0
$113$	153473.	1.13067	0.565335	−	0.824861i	$-0.308747\pi$
$113$	153473.	1.13067	0.565335	+	0.824861i	$0.308747\pi$
$114$	0	0
$115$	415981.	2.93311
$116$	9848.68	0.0679568
$117$	0	0
$118$	184482.	1.21969
$119$	0	0
$120$	0	0
$121$	265761.	1.65016
$122$	46550.5	0.283155
$123$	0	0
$124$	−10685.8	−0.0624095
$125$	461080.	2.63938
$126$	0	0
$127$	142247.	0.782591	0.391296	−	0.920265i	$-0.372027\pi$
$127$	142247.	0.782591	0.391296	+	0.920265i	$0.372027\pi$
$128$	146694.	0.791385
$129$	0	0
$130$	75967.6	0.394249
$131$	127000.	0.646587	0.323293	−	0.946299i	$-0.395210\pi$
$131$	127000.	0.646587	0.323293	+	0.946299i	$0.395210\pi$
$132$	0	0
$133$	0	0
$134$	−64182.2	−0.308782
$135$	0	0
$136$	−223072.	−1.03419
$137$	−99654.8	−0.453625	−0.226812	−	0.973938i	$-0.572830\pi$
$137$	−99654.8	−0.453625	−0.226812	+	0.973938i	$0.572830\pi$
$138$	0	0
$139$	−23441.9	−0.102909	−0.0514547	−	0.998675i	$-0.516386\pi$
$139$	−23441.9	−0.102909	−0.0514547	+	0.998675i	$0.516386\pi$
$140$	0	0
$141$	0	0
$142$	189015.	0.786639
$143$	−90192.2	−0.368832
$144$	0	0
$145$	−274141.	−1.08282
$146$	−178082.	−0.691415
$147$	0	0
$148$	−10619.7	−0.0398526
$149$	−498705.	−1.84025	−0.920127	−	0.391620i	$-0.871915\pi$
$149$	−498705.	−1.84025	−0.920127	+	0.391620i	$0.871915\pi$
$150$	0	0
$151$	−216338.	−0.772130	−0.386065	−	0.922472i	$-0.626166\pi$
$151$	−216338.	−0.772130	−0.386065	+	0.922472i	$0.626166\pi$
$152$	−324961.	−1.14083
$153$	0	0
$154$	0	0
$155$	297441.	0.994426
$156$	0	0
$157$	−426837.	−1.38201	−0.691007	−	0.722848i	$-0.742833\pi$
$157$	−426837.	−1.38201	−0.691007	+	0.722848i	$0.742833\pi$
$158$	229398.	0.731049
$159$	0	0
$160$	138515.	0.427757
$161$	0	0
$162$	0	0
$163$	−273196.	−0.805387	−0.402694	−	0.915335i	$-0.631926\pi$
$163$	−273196.	−0.805387	−0.402694	+	0.915335i	$0.631926\pi$
$164$	−804.735	−0.00233638
$165$	0	0
$166$	−230483.	−0.649184
$167$	600365.	1.66581	0.832903	−	0.553420i	$-0.186677\pi$
$167$	600365.	1.66581	0.832903	+	0.553420i	$0.186677\pi$
$168$	0	0
$169$	−352234.	−0.948668
$170$	646226.	1.71499
$171$	0	0
$172$	−10885.1	−0.0280552
$173$	−500188.	−1.27063	−0.635313	−	0.772254i	$-0.719129\pi$
$173$	−500188.	−1.27063	−0.635313	+	0.772254i	$0.719129\pi$
$174$	0	0
$175$	0	0
$176$	582248.	1.41686
$177$	0	0
$178$	−550606.	−1.30254
$179$	−289563.	−0.675477	−0.337739	−	0.941240i	$-0.609662\pi$
$179$	−289563.	−0.675477	−0.337739	+	0.941240i	$0.609662\pi$
$180$	0	0
$181$	−168243.	−0.381715	−0.190858	−	0.981618i	$-0.561127\pi$
$181$	−168243.	−0.381715	−0.190858	+	0.981618i	$0.561127\pi$
$182$	0	0
$183$	0	0
$184$	−763652.	−1.66284
$185$	295602.	0.635006
$186$	0	0
$187$	−767229.	−1.60443
$188$	−55077.6	−0.113653
$189$	0	0
$190$	941391.	1.89185
$191$	386110.	0.765822	0.382911	−	0.923785i	$-0.374922\pi$
$191$	386110.	0.765822	0.382911	+	0.923785i	$0.374922\pi$
$192$	0	0
$193$	−339682.	−0.656416	−0.328208	−	0.944606i	$-0.606445\pi$
$193$	−339682.	−0.656416	−0.328208	+	0.944606i	$0.606445\pi$
$194$	458927.	0.875466
$195$	0	0
$196$	0	0
$197$	460915.	0.846166	0.423083	−	0.906091i	$-0.360948\pi$
$197$	460915.	0.846166	0.423083	+	0.906091i	$0.360948\pi$
$198$	0	0
$199$	392888.	0.703292	0.351646	−	0.936133i	$-0.385622\pi$
$199$	392888.	0.703292	0.351646	+	0.936133i	$0.385622\pi$
$200$	−1.44004e6	−2.54565
$201$	0	0
$202$	971504.	1.67520
$203$	0	0
$204$	0	0
$205$	22400.0	0.0372275
$206$	213195.	0.350033
$207$	0	0
$208$	−123039.	−0.197189
$209$	−1.11766e6	−1.76988
$210$	0	0
$211$	−469348.	−0.725752	−0.362876	−	0.931837i	$-0.618205\pi$
$211$	−469348.	−0.725752	−0.362876	+	0.931837i	$0.618205\pi$
$212$	78685.4	0.120242
$213$	0	0
$214$	586100.	0.874857
$215$	302991.	0.447027
$216$	0	0
$217$	0	0
$218$	−1.18251e6	−1.68525
$219$	0	0
$220$	251278.	0.350024
$221$	162128.	0.223294
$222$	0	0
$223$	−83059.4	−0.111848	−0.0559238	−	0.998435i	$-0.517810\pi$
$223$	−83059.4	−0.111848	−0.0559238	+	0.998435i	$0.517810\pi$
$224$	0	0
$225$	0	0
$226$	816194.	1.06297
$227$	139769.	0.180030	0.0900152	−	0.995940i	$-0.471308\pi$
$227$	139769.	0.180030	0.0900152	+	0.995940i	$0.471308\pi$
$228$	0	0
$229$	−143847.	−0.181264	−0.0906321	−	0.995884i	$-0.528889\pi$
$229$	−143847.	−0.181264	−0.0906321	+	0.995884i	$0.528889\pi$
$230$	2.21225e6	2.75749
$231$	0	0
$232$	503265.	0.613870
$233$	372492.	0.449498	0.224749	−	0.974417i	$-0.427844\pi$
$233$	372492.	0.449498	0.224749	+	0.974417i	$0.427844\pi$
$234$	0	0
$235$	1.53310e6	1.81093
$236$	−128948.	−0.150707
$237$	0	0
$238$	0	0
$239$	201182.	0.227822	0.113911	−	0.993491i	$-0.463662\pi$
$239$	201182.	0.227822	0.113911	+	0.993491i	$0.463662\pi$
$240$	0	0
$241$	42020.6	0.0466036	0.0233018	−	0.999728i	$-0.492582\pi$
$241$	42020.6	0.0466036	0.0233018	+	0.999728i	$0.492582\pi$
$242$	1.41336e6	1.55136
$243$	0	0
$244$	−32537.4	−0.0349872
$245$	0	0
$246$	0	0
$247$	236180.	0.246321
$248$	−546039.	−0.563760
$249$	0	0
$250$	2.45209e6	2.48135
$251$	−1.23330e6	−1.23562	−0.617809	−	0.786329i	$-0.711979\pi$
$251$	−1.23330e6	−1.23562	−0.617809	+	0.786329i	$0.711979\pi$
$252$	0	0
$253$	−2.62648e6	−2.57972
$254$	756493.	0.735734
$255$	0	0
$256$	−360297.	−0.343606
$257$	560688.	0.529528	0.264764	−	0.964313i	$-0.414706\pi$
$257$	560688.	0.529528	0.264764	+	0.964313i	$0.414706\pi$
$258$	0	0
$259$	0	0
$260$	−53099.2	−0.0487141
$261$	0	0
$262$	675408.	0.607873
$263$	1.44524e6	1.28840	0.644198	−	0.764859i	$-0.277191\pi$
$263$	1.44524e6	1.28840	0.644198	+	0.764859i	$0.277191\pi$
$264$	0	0
$265$	−2.19023e6	−1.91591
$266$	0	0
$267$	0	0
$268$	44861.5	0.0381537
$269$	417542.	0.351819	0.175910	−	0.984406i	$-0.443713\pi$
$269$	417542.	0.351819	0.175910	+	0.984406i	$0.443713\pi$
$270$	0	0
$271$	428512.	0.354437	0.177219	−	0.984172i	$-0.443290\pi$
$271$	428512.	0.354437	0.177219	+	0.984172i	$0.443290\pi$
$272$	−1.04664e6	−0.857778
$273$	0	0
$274$	−529980.	−0.426465
$275$	−4.95283e6	−3.94931
$276$	0	0
$277$	−690887.	−0.541013	−0.270506	−	0.962718i	$-0.587191\pi$
$277$	−690887.	−0.541013	−0.270506	+	0.962718i	$0.587191\pi$
$278$	−124667.	−0.0967478
$279$	0	0
$280$	0	0
$281$	−1.91606e6	−1.44758	−0.723792	−	0.690018i	$-0.757603\pi$
$281$	−1.91606e6	−1.44758	−0.723792	+	0.690018i	$0.757603\pi$
$282$	0	0
$283$	1.62417e6	1.20550	0.602748	−	0.797931i	$-0.294072\pi$
$283$	1.62417e6	1.20550	0.602748	+	0.797931i	$0.294072\pi$
$284$	−132116.	−0.0971986
$285$	0	0
$286$	−479656.	−0.346749
$287$	0	0
$288$	0	0
$289$	−40699.3	−0.0286644
$290$	−1.45793e6	−1.01798
$291$	0	0
$292$	124475.	0.0854326
$293$	−531272.	−0.361533	−0.180767	−	0.983526i	$-0.557858\pi$
$293$	−531272.	−0.361533	−0.180767	+	0.983526i	$0.557858\pi$
$294$	0	0
$295$	3.58930e6	2.40135
$296$	−542661.	−0.359998
$297$	0	0
$298$	−2.65219e6	−1.73007
$299$	555019.	0.359029
$300$	0	0
$301$	0	0
$302$	−1.15052e6	−0.725899
$303$	0	0
$304$	−1.52469e6	−0.946234
$305$	905690.	0.557481
$306$	0	0
$307$	−409059.	−0.247708	−0.123854	−	0.992300i	$-0.539525\pi$
$307$	−409059.	−0.247708	−0.123854	+	0.992300i	$0.539525\pi$
$308$	0	0
$309$	0	0
$310$	1.58184e6	0.934885
$311$	−308704.	−0.180984	−0.0904922	−	0.995897i	$-0.528844\pi$
$311$	−308704.	−0.180984	−0.0904922	+	0.995897i	$0.528844\pi$
$312$	0	0
$313$	−637733.	−0.367941	−0.183970	−	0.982932i	$-0.558895\pi$
$313$	−637733.	−0.367941	−0.183970	+	0.982932i	$0.558895\pi$
$314$	−2.26998e6	−1.29927
$315$	0	0
$316$	−160342.	−0.0903297
$317$	2.94325e6	1.64505	0.822525	−	0.568729i	$-0.192565\pi$
$317$	2.94325e6	1.64505	0.822525	+	0.568729i	$0.192565\pi$
$318$	0	0
$319$	1.73092e6	0.952356
$320$	3.68756e6	2.01309
$321$	0	0
$322$	0	0
$323$	2.00909e6	1.07150
$324$	0	0
$325$	1.04661e6	0.549639
$326$	−1.45290e6	−0.757165
$327$	0	0
$328$	−41121.7	−0.0211051
$329$	0	0
$330$	0	0
$331$	−2.84991e6	−1.42976	−0.714878	−	0.699249i	$-0.753518\pi$
$331$	−2.84991e6	−1.42976	−0.714878	+	0.699249i	$0.753518\pi$
$332$	161101.	0.0802144
$333$	0	0
$334$	3.19283e6	1.56607
$335$	−1.24874e6	−0.607937
$336$	0	0
$337$	3.46617e6	1.66255	0.831277	−	0.555859i	$-0.187611\pi$
$337$	3.46617e6	1.66255	0.831277	+	0.555859i	$0.187611\pi$
$338$	−1.87323e6	−0.891867
$339$	0	0
$340$	−451694.	−0.211908
$341$	−1.87803e6	−0.874615
$342$	0	0
$343$	0	0
$344$	−556228.	−0.253429
$345$	0	0
$346$	−2.66008e6	−1.19455
$347$	−15115.3	−0.00673897	−0.00336949	−	0.999994i	$-0.501073\pi$
$347$	−15115.3	−0.00673897	−0.00336949	+	0.999994i	$0.501073\pi$
$348$	0	0
$349$	−3.24698e6	−1.42698	−0.713488	−	0.700667i	$-0.752886\pi$
$349$	−3.24698e6	−1.42698	−0.713488	+	0.700667i	$0.752886\pi$
$350$	0	0
$351$	0	0
$352$	−874578.	−0.376220
$353$	−1.76786e6	−0.755110	−0.377555	−	0.925987i	$-0.623235\pi$
$353$	−1.76786e6	−0.755110	−0.377555	+	0.925987i	$0.623235\pi$
$354$	0	0
$355$	3.67749e6	1.54875
$356$	384858.	0.160944
$357$	0	0
$358$	−1.53994e6	−0.635034
$359$	121581.	0.0497885	0.0248942	−	0.999690i	$-0.492075\pi$
$359$	121581.	0.0497885	0.0248942	+	0.999690i	$0.492075\pi$
$360$	0	0
$361$	450645.	0.181998
$362$	−894740.	−0.358860
$363$	0	0
$364$	0	0
$365$	−3.46479e6	−1.36127
$366$	0	0
$367$	−2.08841e6	−0.809378	−0.404689	−	0.914454i	$-0.632620\pi$
$367$	−2.08841e6	−0.809378	−0.404689	+	0.914454i	$0.632620\pi$
$368$	−3.58300e6	−1.37920
$369$	0	0
$370$	1.57206e6	0.596985
$371$	0	0
$372$	0	0
$373$	−2.29355e6	−0.853563	−0.426782	−	0.904355i	$-0.640353\pi$
$373$	−2.29355e6	−0.853563	−0.426782	+	0.904355i	$0.640353\pi$
$374$	−4.08024e6	−1.50837
$375$	0	0
$376$	−2.81445e6	−1.02665
$377$	−365771.	−0.132543
$378$	0	0
$379$	−2.93812e6	−1.05068	−0.525341	−	0.850892i	$-0.676062\pi$
$379$	−2.93812e6	−1.05068	−0.525341	+	0.850892i	$0.676062\pi$
$380$	−658005.	−0.233760
$381$	0	0
$382$	2.05339e6	0.719969
$383$	5.04605e6	1.75774	0.878871	−	0.477060i	$-0.158298\pi$
$383$	5.04605e6	1.75774	0.878871	+	0.477060i	$0.158298\pi$
$384$	0	0
$385$	0	0
$386$	−1.80648e6	−0.617113
$387$	0	0
$388$	−320777.	−0.108174
$389$	1.07621e6	0.360599	0.180300	−	0.983612i	$-0.442293\pi$
$389$	1.07621e6	0.360599	0.180300	+	0.983612i	$0.442293\pi$
$390$	0	0
$391$	4.72132e6	1.56179
$392$	0	0
$393$	0	0
$394$	2.45122e6	0.795502
$395$	4.46318e6	1.43930
$396$	0	0
$397$	−4.34345e6	−1.38312	−0.691559	−	0.722320i	$-0.743076\pi$
$397$	−4.34345e6	−1.38312	−0.691559	+	0.722320i	$0.743076\pi$
$398$	2.08944e6	0.661183
$399$	0	0
$400$	−6.75655e6	−2.11142
$401$	704083.	0.218657	0.109328	−	0.994006i	$-0.465130\pi$
$401$	704083.	0.218657	0.109328	+	0.994006i	$0.465130\pi$
$402$	0	0
$403$	396859.	0.121723
$404$	−679054.	−0.206991
$405$	0	0
$406$	0	0
$407$	−1.86642e6	−0.558499
$408$	0	0
$409$	1.63500e6	0.483292	0.241646	−	0.970365i	$-0.422313\pi$
$409$	1.63500e6	0.483292	0.241646	+	0.970365i	$0.422313\pi$
$410$	119127.	0.0349986
$411$	0	0
$412$	−149017.	−0.0432508
$413$	0	0
$414$	0	0
$415$	−4.48429e6	−1.27813
$416$	184813.	0.0523598
$417$	0	0
$418$	−5.94390e6	−1.66391
$419$	−5.92839e6	−1.64969	−0.824844	−	0.565360i	$-0.808737\pi$
$419$	−5.92839e6	−1.64969	−0.824844	+	0.565360i	$0.808737\pi$
$420$	0	0
$421$	−5.46170e6	−1.50184	−0.750918	−	0.660395i	$-0.770389\pi$
$421$	−5.46170e6	−1.50184	−0.750918	+	0.660395i	$0.770389\pi$
$422$	−2.49606e6	−0.682299
$423$	0	0
$424$	4.02080e6	1.08617
$425$	8.90311e6	2.39095
$426$	0	0
$427$	0	0
$428$	−409667.	−0.108099
$429$	0	0
$430$	1.61136e6	0.420262
$431$	−3.44526e6	−0.893366	−0.446683	−	0.894692i	$-0.647395\pi$
$431$	−3.44526e6	−0.893366	−0.446683	+	0.894692i	$0.647395\pi$
$432$	0	0
$433$	−2.46833e6	−0.632680	−0.316340	−	0.948646i	$-0.602454\pi$
$433$	−2.46833e6	−0.632680	−0.316340	+	0.948646i	$0.602454\pi$
$434$	0	0
$435$	0	0
$436$	826541.	0.208232
$437$	6.87779e6	1.72284
$438$	0	0
$439$	6.13780e6	1.52003	0.760014	−	0.649907i	$-0.225192\pi$
$439$	6.13780e6	1.52003	0.760014	+	0.649907i	$0.225192\pi$
$440$	1.28403e7	3.16185
$441$	0	0
$442$	862222.	0.209925
$443$	1.79219e6	0.433885	0.216942	−	0.976184i	$-0.430392\pi$
$443$	1.79219e6	0.433885	0.216942	+	0.976184i	$0.430392\pi$
$444$	0	0
$445$	−1.07126e7	−2.56446
$446$	−441723.	−0.105151
$447$	0	0
$448$	0	0
$449$	3.76014e6	0.880213	0.440107	−	0.897945i	$-0.354941\pi$
$449$	3.76014e6	0.880213	0.440107	+	0.897945i	$0.354941\pi$
$450$	0	0
$451$	−141433.	−0.0327423
$452$	−570496.	−0.131343
$453$	0	0
$454$	743312.	0.169251
$455$	0	0
$456$	0	0
$457$	7.54502e6	1.68994	0.844968	−	0.534817i	$-0.179620\pi$
$457$	7.54502e6	1.68994	0.844968	+	0.534817i	$0.179620\pi$
$458$	−765000.	−0.170411
$459$	0	0
$460$	−1.54630e6	−0.340721
$461$	359098.	0.0786975	0.0393487	−	0.999226i	$-0.487472\pi$
$461$	359098.	0.0786975	0.0393487	+	0.999226i	$0.487472\pi$
$462$	0	0
$463$	−890950.	−0.193153	−0.0965764	−	0.995326i	$-0.530789\pi$
$463$	−890950.	−0.193153	−0.0965764	+	0.995326i	$0.530789\pi$
$464$	2.36128e6	0.509158
$465$	0	0
$466$	1.98097e6	0.422584
$467$	−7.61319e6	−1.61538	−0.807690	−	0.589608i	$-0.799282\pi$
$467$	−7.61319e6	−1.61538	−0.807690	+	0.589608i	$0.799282\pi$
$468$	0	0
$469$	0	0
$470$	8.15328e6	1.70250
$471$	0	0
$472$	−6.58919e6	−1.36137
$473$	−1.91307e6	−0.393169
$474$	0	0
$475$	1.29696e7	2.63751
$476$	0	0
$477$	0	0
$478$	1.06992e6	0.214181
$479$	−5.96007e6	−1.18690	−0.593448	−	0.804872i	$-0.702234\pi$
$479$	−5.96007e6	−1.18690	−0.593448	+	0.804872i	$0.702234\pi$
$480$	0	0
$481$	394404.	0.0777282
$482$	223472.	0.0438132
$483$	0	0
$484$	−987895.	−0.191689
$485$	8.92893e6	1.72363
$486$	0	0
$487$	−257827.	−0.0492612	−0.0246306	−	0.999697i	$-0.507841\pi$
$487$	−257827.	−0.0492612	−0.0246306	+	0.999697i	$0.507841\pi$
$488$	−1.66265e6	−0.316047
$489$	0	0
$490$	0	0
$491$	−592660.	−0.110944	−0.0554718	−	0.998460i	$-0.517666\pi$
$491$	−592660.	−0.110944	−0.0554718	+	0.998460i	$0.517666\pi$
$492$	0	0
$493$	−3.11146e6	−0.576564
$494$	1.25604e6	0.231573
$495$	0	0
$496$	−2.56198e6	−0.467596
$497$	0	0
$498$	0	0
$499$	−2.42242e6	−0.435511	−0.217755	−	0.976003i	$-0.569874\pi$
$499$	−2.42242e6	−0.435511	−0.217755	+	0.976003i	$0.569874\pi$
$500$	−1.71394e6	−0.306600
$501$	0	0
$502$	−6.55887e6	−1.16164
$503$	2.31849e6	0.408588	0.204294	−	0.978910i	$-0.434510\pi$
$503$	2.31849e6	0.408588	0.204294	+	0.978910i	$0.434510\pi$
$504$	0	0
$505$	1.89017e7	3.29816
$506$	−1.39680e7	−2.42527
$507$	0	0
$508$	−528767.	−0.0909087
$509$	−8.49886e6	−1.45401	−0.727003	−	0.686634i	$-0.759087\pi$
$509$	−8.49886e6	−1.45401	−0.727003	+	0.686634i	$0.759087\pi$
$510$	0	0
$511$	0	0
$512$	−6.61033e6	−1.11442
$513$	0	0
$514$	2.98183e6	0.497822
$515$	4.14795e6	0.689152
$516$	0	0
$517$	−9.67995e6	−1.59275
$518$	0	0
$519$	0	0
$520$	−2.71335e6	−0.440046
$521$	−1.16092e6	−0.187374	−0.0936868	−	0.995602i	$-0.529865\pi$
$521$	−1.16092e6	−0.187374	−0.0936868	+	0.995602i	$0.529865\pi$
$522$	0	0
$523$	−1.22907e6	−0.196482	−0.0982411	−	0.995163i	$-0.531322\pi$
$523$	−1.22907e6	−0.196482	−0.0982411	+	0.995163i	$0.531322\pi$
$524$	−472091.	−0.0751099
$525$	0	0
$526$	7.68598e6	1.21125
$527$	3.37592e6	0.529499
$528$	0	0
$529$	9.72633e6	1.51116
$530$	−1.16480e7	−1.80120
$531$	0	0
$532$	0	0
$533$	29887.1	0.00455686
$534$	0	0
$535$	1.14032e7	1.72244
$536$	2.29241e6	0.344652
$537$	0	0
$538$	2.22055e6	0.330754
$539$	0	0
$540$	0	0
$541$	−790279.	−0.116088	−0.0580440	−	0.998314i	$-0.518486\pi$
$541$	−790279.	−0.116088	−0.0580440	+	0.998314i	$0.518486\pi$
$542$	2.27889e6	0.333216
$543$	0	0
$544$	1.57213e6	0.227767
$545$	−2.30070e7	−3.31795
$546$	0	0
$547$	1.74335e6	0.249124	0.124562	−	0.992212i	$-0.460247\pi$
$547$	1.74335e6	0.249124	0.124562	+	0.992212i	$0.460247\pi$
$548$	370441.	0.0526947
$549$	0	0
$550$	−2.63399e7	−3.71285
$551$	−4.53263e6	−0.636021
$552$	0	0
$553$	0	0
$554$	−3.67424e6	−0.508620
$555$	0	0
$556$	87139.0	0.0119543
$557$	1.01834e7	1.39077	0.695385	−	0.718637i	$-0.255234\pi$
$557$	1.01834e7	1.39077	0.695385	+	0.718637i	$0.255234\pi$
$558$	0	0
$559$	404264.	0.0547186
$560$	0	0
$561$	0	0
$562$	−1.01899e7	−1.36091
$563$	−8.57420e6	−1.14005	−0.570023	−	0.821629i	$-0.693066\pi$
$563$	−8.57420e6	−1.14005	−0.570023	+	0.821629i	$0.693066\pi$
$564$	0	0
$565$	1.58799e7	2.09280
$566$	8.63760e6	1.13332
$567$	0	0
$568$	−6.75109e6	−0.878018
$569$	−2.67960e6	−0.346968	−0.173484	−	0.984837i	$-0.555502\pi$
$569$	−2.67960e6	−0.346968	−0.173484	+	0.984837i	$0.555502\pi$
$570$	0	0
$571$	7.45017e6	0.956260	0.478130	−	0.878289i	$-0.341315\pi$
$571$	7.45017e6	0.956260	0.478130	+	0.878289i	$0.341315\pi$
$572$	335266.	0.0428449
$573$	0	0
$574$	0	0
$575$	3.04784e7	3.84434
$576$	0	0
$577$	−5.72891e6	−0.716362	−0.358181	−	0.933652i	$-0.616603\pi$
$577$	−5.72891e6	−0.716362	−0.358181	+	0.933652i	$0.616603\pi$
$578$	−216445.	−0.0269481
$579$	0	0
$580$	1.01905e6	0.125784
$581$	0	0
$582$	0	0
$583$	1.38290e7	1.68508
$584$	6.36061e6	0.771733
$585$	0	0
$586$	−2.82539e6	−0.339887
$587$	1.41744e7	1.69788	0.848942	−	0.528486i	$-0.177240\pi$
$587$	1.41744e7	1.69788	0.848942	+	0.528486i	$0.177240\pi$
$588$	0	0
$589$	4.91787e6	0.584102
$590$	1.90885e7	2.25757
$591$	0	0
$592$	−2.54613e6	−0.298591
$593$	−6.09494e6	−0.711758	−0.355879	−	0.934532i	$-0.615818\pi$
$593$	−6.09494e6	−0.711758	−0.355879	+	0.934532i	$0.615818\pi$
$594$	0	0
$595$	0	0
$596$	1.85380e6	0.213771
$597$	0	0
$598$	2.95168e6	0.337533
$599$	−1.97464e6	−0.224865	−0.112432	−	0.993659i	$-0.535864\pi$
$599$	−1.97464e6	−0.224865	−0.112432	+	0.993659i	$0.535864\pi$
$600$	0	0
$601$	−5.14078e6	−0.580554	−0.290277	−	0.956943i	$-0.593747\pi$
$601$	−5.14078e6	−0.580554	−0.290277	+	0.956943i	$0.593747\pi$
$602$	0	0
$603$	0	0
$604$	804180.	0.0896935
$605$	2.74984e7	3.05435
$606$	0	0
$607$	1.31482e6	0.144843	0.0724213	−	0.997374i	$-0.476927\pi$
$607$	1.31482e6	0.144843	0.0724213	+	0.997374i	$0.476927\pi$
$608$	2.29020e6	0.251255
$609$	0	0
$610$	4.81660e6	0.524102
$611$	2.04553e6	0.221668
$612$	0	0
$613$	1.34181e7	1.44224	0.721122	−	0.692808i	$-0.243627\pi$
$613$	1.34181e7	1.44224	0.721122	+	0.692808i	$0.243627\pi$
$614$	−2.17544e6	−0.232877
$615$	0	0
$616$	0	0
$617$	−1.48394e7	−1.56929	−0.784646	−	0.619944i	$-0.787155\pi$
$617$	−1.48394e7	−1.56929	−0.784646	+	0.619944i	$0.787155\pi$
$618$	0	0
$619$	1.14468e7	1.20077	0.600383	−	0.799713i	$-0.295015\pi$
$619$	1.14468e7	1.20077	0.600383	+	0.799713i	$0.295015\pi$
$620$	−1.10566e6	−0.115516
$621$	0	0
$622$	−1.64173e6	−0.170148
$623$	0	0
$624$	0	0
$625$	2.40171e7	2.45935
$626$	−3.39156e6	−0.345910
$627$	0	0
$628$	1.58665e6	0.160540
$629$	3.35504e6	0.338120
$630$	0	0
$631$	−1.11608e7	−1.11589	−0.557944	−	0.829878i	$-0.688410\pi$
$631$	−1.11608e7	−1.11589	−0.557944	+	0.829878i	$0.688410\pi$
$632$	−8.19345e6	−0.815970
$633$	0	0
$634$	1.56527e7	1.54655
$635$	1.47184e7	1.44853
$636$	0	0
$637$	0	0
$638$	9.20528e6	0.895334
$639$	0	0
$640$	1.51785e7	1.46480
$641$	1.72150e7	1.65486	0.827431	−	0.561567i	$-0.189801\pi$
$641$	1.72150e7	1.65486	0.827431	+	0.561567i	$0.189801\pi$
$642$	0	0
$643$	2.73062e6	0.260456	0.130228	−	0.991484i	$-0.458429\pi$
$643$	2.73062e6	0.260456	0.130228	+	0.991484i	$0.458429\pi$
$644$	0	0
$645$	0	0
$646$	1.06846e7	1.00735
$647$	−4.39969e6	−0.413201	−0.206600	−	0.978425i	$-0.566240\pi$
$647$	−4.39969e6	−0.413201	−0.206600	+	0.978425i	$0.566240\pi$
$648$	0	0
$649$	−2.26627e7	−2.11203
$650$	5.56605e6	0.516730
$651$	0	0
$652$	1.01553e6	0.0935567
$653$	−5.60500e6	−0.514390	−0.257195	−	0.966360i	$-0.582798\pi$
$653$	−5.60500e6	−0.514390	−0.257195	+	0.966360i	$0.582798\pi$
$654$	0	0
$655$	1.31408e7	1.19679
$656$	−192940.	−0.0175050
$657$	0	0
$658$	0	0
$659$	1.33959e7	1.20159	0.600796	−	0.799402i	$-0.294850\pi$
$659$	1.33959e7	1.20159	0.600796	+	0.799402i	$0.294850\pi$
$660$	0	0
$661$	−1.87657e6	−0.167056	−0.0835279	−	0.996505i	$-0.526619\pi$
$661$	−1.87657e6	−0.167056	−0.0835279	+	0.996505i	$0.526619\pi$
$662$	−1.51563e7	−1.34415
$663$	0	0
$664$	8.23220e6	0.724596
$665$	0	0
$666$	0	0
$667$	−1.06516e7	−0.927043
$668$	−2.23170e6	−0.193506
$669$	0	0
$670$	−6.64097e6	−0.571537
$671$	−5.71849e6	−0.490315
$672$	0	0
$673$	2.82367e6	0.240313	0.120156	−	0.992755i	$-0.461660\pi$
$673$	2.82367e6	0.240313	0.120156	+	0.992755i	$0.461660\pi$
$674$	1.84336e7	1.56301
$675$	0	0
$676$	1.30934e6	0.110201
$677$	9.31849e6	0.781401	0.390700	−	0.920518i	$-0.372233\pi$
$677$	9.31849e6	0.781401	0.390700	+	0.920518i	$0.372233\pi$
$678$	0	0
$679$	0	0
$680$	−2.30814e7	−1.91421
$681$	0	0
$682$	−9.98766e6	−0.822248
$683$	2.13619e7	1.75222	0.876108	−	0.482114i	$-0.160131\pi$
$683$	2.13619e7	1.75222	0.876108	+	0.482114i	$0.160131\pi$
$684$	0	0
$685$	−1.03113e7	−0.839631
$686$	0	0
$687$	0	0
$688$	−2.60978e6	−0.210200
$689$	−2.92230e6	−0.234518
$690$	0	0
$691$	−1.10885e7	−0.883439	−0.441720	−	0.897153i	$-0.645631\pi$
$691$	−1.10885e7	−0.883439	−0.441720	+	0.897153i	$0.645631\pi$
$692$	1.85932e6	0.147601
$693$	0	0
$694$	−80385.6	−0.00633548
$695$	−2.42554e6	−0.190479
$696$	0	0
$697$	254237.	0.0198224
$698$	−1.72680e7	−1.34154
$699$	0	0
$700$	0	0
$701$	2.02256e7	1.55456	0.777279	−	0.629156i	$-0.216599\pi$
$701$	2.02256e7	1.55456	0.777279	+	0.629156i	$0.216599\pi$
$702$	0	0
$703$	4.88745e6	0.372988
$704$	−2.32831e7	−1.77055
$705$	0	0
$706$	−9.40174e6	−0.709899
$707$	0	0
$708$	0	0
$709$	−2.10834e7	−1.57516	−0.787580	−	0.616213i	$-0.788666\pi$
$709$	−2.10834e7	−1.57516	−0.787580	+	0.616213i	$0.788666\pi$
$710$	1.95575e7	1.45602
$711$	0	0
$712$	1.96661e7	1.45385
$713$	1.15569e7	0.851369
$714$	0	0
$715$	−9.33224e6	−0.682686
$716$	1.07638e6	0.0784659
$717$	0	0
$718$	646585.	0.0468074
$719$	−5.88060e6	−0.424228	−0.212114	−	0.977245i	$-0.568035\pi$
$719$	−5.88060e6	−0.424228	−0.212114	+	0.977245i	$0.568035\pi$
$720$	0	0
$721$	0	0
$722$	2.39660e6	0.171101
$723$	0	0
$724$	625398.	0.0443414
$725$	−2.00860e7	−1.41921
$726$	0	0
$727$	−9.23778e6	−0.648233	−0.324117	−	0.946017i	$-0.605067\pi$
$727$	−9.23778e6	−0.648233	−0.324117	+	0.946017i	$0.605067\pi$
$728$	0	0
$729$	0	0
$730$	−1.84263e7	−1.27977
$731$	3.43891e6	0.238028
$732$	0	0
$733$	−7.75113e6	−0.532850	−0.266425	−	0.963856i	$-0.585842\pi$
$733$	−7.75113e6	−0.532850	−0.266425	+	0.963856i	$0.585842\pi$
$734$	−1.11065e7	−0.760917
$735$	0	0
$736$	5.38192e6	0.366220
$737$	7.88446e6	0.534691
$738$	0	0
$739$	1.02902e7	0.693125	0.346563	−	0.938027i	$-0.387349\pi$
$739$	1.02902e7	0.693125	0.346563	+	0.938027i	$0.387349\pi$
$740$	−1.09882e6	−0.0737646
$741$	0	0
$742$	0	0
$743$	2.04494e6	0.135897	0.0679483	−	0.997689i	$-0.478355\pi$
$743$	2.04494e6	0.135897	0.0679483	+	0.997689i	$0.478355\pi$
$744$	0	0
$745$	−5.16012e7	−3.40619
$746$	−1.21974e7	−0.802457
$747$	0	0
$748$	2.85197e6	0.186377
$749$	0	0
$750$	0	0
$751$	−1.18742e7	−0.768251	−0.384125	−	0.923281i	$-0.625497\pi$
$751$	−1.18742e7	−0.768251	−0.384125	+	0.923281i	$0.625497\pi$
$752$	−1.32052e7	−0.851531
$753$	0	0
$754$	−1.94523e6	−0.124607
$755$	−2.23846e7	−1.42916
$756$	0	0
$757$	−8.48702e6	−0.538289	−0.269145	−	0.963100i	$-0.586741\pi$
$757$	−8.48702e6	−0.538289	−0.269145	+	0.963100i	$0.586741\pi$
$758$	−1.56254e7	−0.987772
$759$	0	0
$760$	−3.36239e7	−2.11161
$761$	2.39932e7	1.50185	0.750925	−	0.660388i	$-0.229608\pi$
$761$	2.39932e7	1.50185	0.750925	+	0.660388i	$0.229608\pi$
$762$	0	0
$763$	0	0
$764$	−1.43526e6	−0.0889607
$765$	0	0
$766$	2.68357e7	1.65250
$767$	4.78899e6	0.293938
$768$	0	0
$769$	2.43194e7	1.48299	0.741494	−	0.670959i	$-0.234117\pi$
$769$	2.43194e7	1.48299	0.741494	+	0.670959i	$0.234117\pi$
$770$	0	0
$771$	0	0
$772$	1.26268e6	0.0762517
$773$	−5.38149e6	−0.323932	−0.161966	−	0.986796i	$-0.551784\pi$
$773$	−5.38149e6	−0.323932	−0.161966	+	0.986796i	$0.551784\pi$
$774$	0	0
$775$	2.17931e7	1.30336
$776$	−1.63916e7	−0.977164
$777$	0	0
$778$	5.72347e6	0.339009
$779$	370361.	0.0218666
$780$	0	0
$781$	−2.32195e7	−1.36215
$782$	2.51087e7	1.46828
$783$	0	0
$784$	0	0
$785$	−4.41650e7	−2.55802
$786$	0	0
$787$	−2.80718e7	−1.61560	−0.807800	−	0.589456i	$-0.799342\pi$
$787$	−2.80718e7	−1.61560	−0.807800	+	0.589456i	$0.799342\pi$
$788$	−1.71333e6	−0.0982938
$789$	0	0
$790$	2.37359e7	1.35312
$791$	0	0
$792$	0	0
$793$	1.20841e6	0.0682388
$794$	−2.30992e7	−1.30030
$795$	0	0
$796$	−1.46046e6	−0.0816970
$797$	1.97055e7	1.09886	0.549428	−	0.835541i	$-0.314846\pi$
$797$	1.97055e7	1.09886	0.549428	+	0.835541i	$0.314846\pi$
$798$	0	0
$799$	1.74005e7	0.964262
$800$	1.01488e7	0.560648
$801$	0	0
$802$	3.74442e6	0.205565
$803$	2.18765e7	1.19726
$804$	0	0
$805$	0	0
$806$	2.11056e6	0.114435
$807$	0	0
$808$	−3.46994e7	−1.86980
$809$	3.39408e6	0.182327	0.0911633	−	0.995836i	$-0.470941\pi$
$809$	3.39408e6	0.182327	0.0911633	+	0.995836i	$0.470941\pi$
$810$	0	0
$811$	−1.78837e7	−0.954786	−0.477393	−	0.878690i	$-0.658418\pi$
$811$	−1.78837e7	−0.954786	−0.477393	+	0.878690i	$0.658418\pi$
$812$	0	0
$813$	0	0
$814$	−9.92588e6	−0.525059
$815$	−2.82677e7	−1.49072
$816$	0	0
$817$	5.00964e6	0.262574
$818$	8.69518e6	0.454355
$819$	0	0
$820$	−83266.3	−0.00432449
$821$	−9.26949e6	−0.479952	−0.239976	−	0.970779i	$-0.577140\pi$
$821$	−9.26949e6	−0.479952	−0.239976	+	0.970779i	$0.577140\pi$
$822$	0	0
$823$	−1.52579e6	−0.0785228	−0.0392614	−	0.999229i	$-0.512501\pi$
$823$	−1.52579e6	−0.0785228	−0.0392614	+	0.999229i	$0.512501\pi$
$824$	−7.61474e6	−0.390694
$825$	0	0
$826$	0	0
$827$	−806595.	−0.0410102	−0.0205051	−	0.999790i	$-0.506527\pi$
$827$	−806595.	−0.0410102	−0.0205051	+	0.999790i	$0.506527\pi$
$828$	0	0
$829$	2.19275e7	1.10816	0.554080	−	0.832463i	$-0.313070\pi$
$829$	2.19275e7	1.10816	0.554080	+	0.832463i	$0.313070\pi$
$830$	−2.38482e7	−1.20160
$831$	0	0
$832$	4.92010e6	0.246414
$833$	0	0
$834$	0	0
$835$	6.21201e7	3.08330
$836$	4.15462e6	0.205596
$837$	0	0
$838$	−3.15281e7	−1.55091
$839$	−2.40115e7	−1.17765	−0.588823	−	0.808262i	$-0.700409\pi$
$839$	−2.40115e7	−1.17765	−0.588823	+	0.808262i	$0.700409\pi$
$840$	0	0
$841$	−1.34915e7	−0.657764
$842$	−2.90462e7	−1.41192
$843$	0	0
$844$	1.74468e6	0.0843061
$845$	−3.64458e7	−1.75593
$846$	0	0
$847$	0	0
$848$	1.88653e7	0.900895
$849$	0	0
$850$	4.73481e7	2.24779
$851$	1.14854e7	0.543654
$852$	0	0
$853$	−2.26785e7	−1.06719	−0.533594	−	0.845741i	$-0.679159\pi$
$853$	−2.26785e7	−1.06719	−0.533594	+	0.845741i	$0.679159\pi$
$854$	0	0
$855$	0	0
$856$	−2.09339e7	−0.976484
$857$	5.07518e6	0.236048	0.118024	−	0.993011i	$-0.462344\pi$
$857$	5.07518e6	0.236048	0.118024	+	0.993011i	$0.462344\pi$
$858$	0	0
$859$	1.71405e7	0.792577	0.396289	−	0.918126i	$-0.370298\pi$
$859$	1.71405e7	0.792577	0.396289	+	0.918126i	$0.370298\pi$
$860$	−1.12629e6	−0.0519284
$861$	0	0
$862$	−1.83224e7	−0.839876
$863$	−3.84686e7	−1.75824	−0.879122	−	0.476597i	$-0.841870\pi$
$863$	−3.84686e7	−1.75824	−0.879122	+	0.476597i	$0.841870\pi$
$864$	0	0
$865$	−5.17547e7	−2.35185
$866$	−1.31270e7	−0.594799
$867$	0	0
$868$	0	0
$869$	−2.81803e7	−1.26589
$870$	0	0
$871$	−1.66611e6	−0.0744148
$872$	4.22360e7	1.88101
$873$	0	0
$874$	3.65772e7	1.61969
$875$	0	0
$876$	0	0
$877$	−1.91671e7	−0.841504	−0.420752	−	0.907176i	$-0.638234\pi$
$877$	−1.91671e7	−0.841504	−0.420752	+	0.907176i	$0.638234\pi$
$878$	3.26418e7	1.42902
$879$	0	0
$880$	6.02455e7	2.62252
$881$	4.15340e7	1.80287	0.901434	−	0.432916i	$-0.142515\pi$
$881$	4.15340e7	1.80287	0.901434	+	0.432916i	$0.142515\pi$
$882$	0	0
$883$	−4.31950e7	−1.86437	−0.932183	−	0.361987i	$-0.882099\pi$
$883$	−4.31950e7	−1.86437	−0.932183	+	0.361987i	$0.882099\pi$
$884$	−602668.	−0.0259387
$885$	0	0
$886$	9.53113e6	0.407906
$887$	6.87508e6	0.293406	0.146703	−	0.989181i	$-0.453134\pi$
$887$	6.87508e6	0.293406	0.146703	+	0.989181i	$0.453134\pi$
$888$	0	0
$889$	0	0
$890$	−5.69715e7	−2.41092
$891$	0	0
$892$	308752.	0.0129926
$893$	2.53482e7	1.06370
$894$	0	0
$895$	−2.99613e7	−1.25027
$896$	0	0
$897$	0	0
$898$	1.99970e7	0.827511
$899$	−7.61628e6	−0.314299
$900$	0	0
$901$	−2.48588e7	−1.02016
$902$	−752162.	−0.0307819
$903$	0	0
$904$	−2.91522e7	−1.18645
$905$	−1.74081e7	−0.706531
$906$	0	0
$907$	1.49747e7	0.604423	0.302212	−	0.953241i	$-0.402275\pi$
$907$	1.49747e7	0.604423	0.302212	+	0.953241i	$0.402275\pi$
$908$	−519554.	−0.0209130
$909$	0	0
$910$	0	0
$911$	−3.13061e7	−1.24978	−0.624889	−	0.780714i	$-0.714856\pi$
$911$	−3.13061e7	−1.24978	−0.624889	+	0.780714i	$0.714856\pi$
$912$	0	0
$913$	2.83136e7	1.12414
$914$	4.01256e7	1.58875
$915$	0	0
$916$	534713.	0.0210563
$917$	0	0
$918$	0	0
$919$	1.46990e7	0.574117	0.287058	−	0.957913i	$-0.407323\pi$
$919$	1.46990e7	0.574117	0.287058	+	0.957913i	$0.407323\pi$
$920$	−7.90154e7	−3.07781
$921$	0	0
$922$	1.90974e6	0.0739855
$923$	4.90666e6	0.189575
$924$	0	0
$925$	2.16583e7	0.832283
$926$	−4.73821e6	−0.181588
$927$	0	0
$928$	−3.54681e6	−0.135197
$929$	1.68420e7	0.640256	0.320128	−	0.947374i	$-0.396274\pi$
$929$	1.68420e7	0.640256	0.320128	+	0.947374i	$0.396274\pi$
$930$	0	0
$931$	0	0
$932$	−1.38464e6	−0.0522153
$933$	0	0
$934$	−4.04881e7	−1.51866
$935$	−7.93856e7	−2.96970
$936$	0	0
$937$	3.22026e7	1.19824	0.599118	−	0.800661i	$-0.295518\pi$
$937$	3.22026e7	1.19824	0.599118	+	0.800661i	$0.295518\pi$
$938$	0	0
$939$	0	0
$940$	−5.69891e6	−0.210364
$941$	2.65064e7	0.975837	0.487919	−	0.872889i	$-0.337756\pi$
$941$	2.65064e7	0.975837	0.487919	+	0.872889i	$0.337756\pi$
$942$	0	0
$943$	870340.	0.0318720
$944$	−3.09160e7	−1.12915
$945$	0	0
$946$	−1.01740e7	−0.369628
$947$	4.98682e7	1.80696	0.903481	−	0.428629i	$-0.141003\pi$
$947$	4.98682e7	1.80696	0.903481	+	0.428629i	$0.141003\pi$
$948$	0	0
$949$	−4.62287e6	−0.166627
$950$	6.89745e7	2.47959
$951$	0	0
$952$	0	0
$953$	2.46027e7	0.877508	0.438754	−	0.898607i	$-0.355420\pi$
$953$	2.46027e7	0.877508	0.438754	+	0.898607i	$0.355420\pi$
$954$	0	0
$955$	3.99510e7	1.41749
$956$	−747843.	−0.0264646
$957$	0	0
$958$	−3.16966e7	−1.11583
$959$	0	0
$960$	0	0
$961$	−2.03655e7	−0.711357
$962$	2.09750e6	0.0730743
$963$	0	0
$964$	−156200.	−0.00541364
$965$	−3.51471e7	−1.21498
$966$	0	0
$967$	1.59962e7	0.550111	0.275056	−	0.961428i	$-0.411304\pi$
$967$	1.59962e7	0.550111	0.275056	+	0.961428i	$0.411304\pi$
$968$	−5.04812e7	−1.73157
$969$	0	0
$970$	4.74854e7	1.62043
$971$	9.94424e6	0.338473	0.169236	−	0.985576i	$-0.445870\pi$
$971$	9.94424e6	0.338473	0.169236	+	0.985576i	$0.445870\pi$
$972$	0	0
$973$	0	0
$974$	−1.37116e6	−0.0463118
$975$	0	0
$976$	−7.80105e6	−0.262137
$977$	1.20871e7	0.405121	0.202561	−	0.979270i	$-0.435074\pi$
$977$	1.20871e7	0.405121	0.202561	+	0.979270i	$0.435074\pi$
$978$	0	0
$979$	6.76392e7	2.25549
$980$	0	0
$981$	0	0
$982$	−3.15186e6	−0.104301
$983$	3.42776e7	1.13143	0.565714	−	0.824601i	$-0.308601\pi$
$983$	3.42776e7	1.13143	0.565714	+	0.824601i	$0.308601\pi$
$984$	0	0
$985$	4.76911e7	1.56620
$986$	−1.65472e7	−0.542043
$987$	0	0
$988$	−877938.	−0.0286135
$989$	1.17725e7	0.382719
$990$	0	0
$991$	−3.50099e7	−1.13242	−0.566209	−	0.824262i	$-0.691590\pi$
$991$	−3.50099e7	−1.13242	−0.566209	+	0.824262i	$0.691590\pi$
$992$	3.84827e6	0.124161
$993$	0	0
$994$	0	0
$995$	4.06523e7	1.30175
$996$	0	0
$997$	−9.99230e6	−0.318367	−0.159183	−	0.987249i	$-0.550886\pi$
$997$	−9.99230e6	−0.318367	−0.159183	+	0.987249i	$0.550886\pi$
$998$	−1.28828e7	−0.409435
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.6.a.bb.1.5		6
3.2	odd	2		147.6.a.n.1.2	✓	6
7.6	odd	2		441.6.a.ba.1.5		6
21.2	odd	6		147.6.e.q.67.5		12
21.5	even	6		147.6.e.p.67.5		12
21.11	odd	6		147.6.e.q.79.5		12
21.17	even	6		147.6.e.p.79.5		12
21.20	even	2		147.6.a.o.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
147.6.a.n.1.2	✓	6	3.2	odd	2
147.6.a.o.1.2	yes	6	21.20	even	2
147.6.e.p.67.5		12	21.5	even	6
147.6.e.p.79.5		12	21.17	even	6
147.6.e.q.67.5		12	21.2	odd	6
147.6.e.q.79.5		12	21.11	odd	6
441.6.a.ba.1.5		6	7.6	odd	2
441.6.a.bb.1.5		6	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

\(f(q)\)	\(=\)	\(q+5.31815 q^{2} -3.71724 q^{4} +103.471 q^{5} -189.950 q^{8} +O(q^{10})\)	\(q+5.31815 q^{2} -3.71724 q^{4} +103.471 q^{5} -189.950 q^{8} +550.272 q^{10} -653.308 q^{11} +138.055 q^{13} -891.231 q^{16} +1174.38 q^{17} +1710.77 q^{19} -384.625 q^{20} -3474.39 q^{22} +4020.28 q^{23} +7581.15 q^{25} +734.196 q^{26} -2649.46 q^{29} +2874.65 q^{31} +1338.69 q^{32} +6245.51 q^{34} +2856.87 q^{37} +9098.15 q^{38} -19654.2 q^{40} +216.487 q^{41} +2928.29 q^{43} +2428.50 q^{44} +21380.5 q^{46} +14816.8 q^{47} +40317.7 q^{50} -513.182 q^{52} -21167.7 q^{53} -67598.1 q^{55} -14090.3 q^{58} +34689.1 q^{59} +8753.12 q^{61} +15287.8 q^{62} +35638.7 q^{64} +14284.6 q^{65} -12068.5 q^{67} -4365.43 q^{68} +35541.5 q^{71} -33485.8 q^{73} +15193.3 q^{74} -6359.35 q^{76} +43134.8 q^{79} -92216.1 q^{80} +1151.31 q^{82} -43338.8 q^{83} +121513. q^{85} +15573.1 q^{86} +124096. q^{88} -103533. q^{89} -14944.3 q^{92} +78798.1 q^{94} +177015. q^{95} +86294.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) \) 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8	\( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97}+O(q^{100}) \) 6 * q - 2 * q^2 + 150 * q^4 + 100 * q^5 + 114 * q^8 - 864 * q^10 - 604 * q^11 - 1352 * q^13 + 4578 * q^16 + 3028 * q^17 - 1728 * q^19 + 452 * q^20 - 4116 * q^22 + 4484 * q^23 + 4806 * q^25 + 14172 * q^26 + 5320 * q^29 - 3976 * q^31 + 37326 * q^32 + 16336 * q^34 + 22680 * q^37 + 52744 * q^38 - 100600 * q^40 + 28756 * q^41 - 6768 * q^43 + 64940 * q^44 + 540 * q^46 + 51552 * q^47 + 40622 * q^50 - 119296 * q^52 - 80884 * q^53 - 11656 * q^55 - 70464 * q^58 + 8872 * q^59 - 50896 * q^61 + 11824 * q^62 + 199590 * q^64 - 3492 * q^65 + 6480 * q^67 + 37348 * q^68 + 110852 * q^71 - 64232 * q^73 + 27464 * q^74 + 194864 * q^76 + 111696 * q^79 - 308940 * q^80 + 189640 * q^82 + 101128 * q^83 - 23292 * q^85 - 3824 * q^86 - 97788 * q^88 - 35012 * q^89 + 449260 * q^92 + 121016 * q^94 + 119080 * q^95 - 70952 * q^97

Introduction

L-functions

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