LMFDB - Embedded newform 448.6.a.y.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,6,Mod(1,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,14,0,-34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$71.8519512762$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{61}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 15 $ x^2 - x - 15
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 224)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.40512$ of defining polynomial
Character	$\chi$	$=$	448.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.810250 q^{3} -71.6717 q^{5} -49.0000 q^{7} -242.343 q^{9} -569.271 q^{11} -137.702 q^{13} +58.0720 q^{15} -418.657 q^{17} -2552.13 q^{19} +39.7022 q^{21} -127.036 q^{23} +2011.84 q^{25} +393.249 q^{27} -2312.40 q^{29} -3992.18 q^{31} +461.252 q^{33} +3511.92 q^{35} +3853.34 q^{37} +111.573 q^{39} -4940.82 q^{41} -13679.2 q^{43} +17369.2 q^{45} +27624.7 q^{47} +2401.00 q^{49} +339.216 q^{51} -37400.0 q^{53} +40800.7 q^{55} +2067.86 q^{57} -37000.0 q^{59} +3803.32 q^{61} +11874.8 q^{63} +9869.36 q^{65} +22454.7 q^{67} +102.931 q^{69} +55088.9 q^{71} -69256.6 q^{73} -1630.09 q^{75} +27894.3 q^{77} +40937.8 q^{79} +58570.8 q^{81} +19789.0 q^{83} +30005.8 q^{85} +1873.62 q^{87} +104148. q^{89} +6747.41 q^{91} +3234.66 q^{93} +182916. q^{95} -96649.8 q^{97} +137959. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{3} - 34 q^{5} - 98 q^{7} - 266 q^{9} - 420 q^{11} + 490 q^{13} + 616 q^{15} - 1056 q^{17} - 1246 q^{19} - 686 q^{21} - 504 q^{23} + 306 q^{25} - 3556 q^{27} + 3904 q^{29} + 2044 q^{31} + 2672 q^{33}+ \cdots + 134428 q^{99}+O(q^{100}) $ 2 * q + 14 * q^3 - 34 * q^5 - 98 * q^7 - 266 * q^9 - 420 * q^11 + 490 * q^13 + 616 * q^15 - 1056 * q^17 - 1246 * q^19 - 686 * q^21 - 504 * q^23 + 306 * q^25 - 3556 * q^27 + 3904 * q^29 + 2044 * q^31 + 2672 * q^33 + 1666 * q^35 + 7488 * q^37 + 9408 * q^39 + 7832 * q^41 - 10332 * q^43 + 16478 * q^45 + 41972 * q^47 + 4802 * q^49 - 9100 * q^51 - 32812 * q^53 + 46424 * q^55 + 21412 * q^57 - 48398 * q^59 + 718 * q^61 + 13034 * q^63 + 33516 * q^65 - 12824 * q^67 - 5480 * q^69 + 103992 * q^71 - 54100 * q^73 - 26894 * q^75 + 20580 * q^77 + 64568 * q^79 + 5830 * q^81 + 47810 * q^83 + 5996 * q^85 + 93940 * q^87 - 17388 * q^89 - 24010 * q^91 + 92632 * q^93 + 232120 * q^95 - 97296 * q^97 + 134428 * q^99

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
$3$	−0.810250	−0.0519775	−0.0259888	−	0.999662i	$-0.508273\pi$
$3$	−0.810250	−0.0519775	−0.0259888	+	0.999662i	$0.508273\pi$
$4$	0	0
$5$	−71.6717	−1.28210	−0.641052	−	0.767498i	$-0.721502\pi$
$5$	−71.6717	−1.28210	−0.641052	+	0.767498i	$0.721502\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−242.343	−0.997298
$10$	0	0
$11$	−569.271	−1.41853	−0.709264	−	0.704943i	$-0.750972\pi$
$11$	−569.271	−1.41853	−0.709264	+	0.704943i	$0.750972\pi$
$12$	0	0
$13$	−137.702	−0.225987	−0.112993	−	0.993596i	$-0.536044\pi$
$13$	−137.702	−0.225987	−0.112993	+	0.993596i	$0.536044\pi$
$14$	0	0
$15$	58.0720	0.0666406
$16$	0	0
$17$	−418.657	−0.351346	−0.175673	−	0.984449i	$-0.556210\pi$
$17$	−418.657	−0.351346	−0.175673	+	0.984449i	$0.556210\pi$
$18$	0	0
$19$	−2552.13	−1.62188	−0.810941	−	0.585128i	$-0.801044\pi$
$19$	−2552.13	−1.62188	−0.810941	+	0.585128i	$0.801044\pi$
$20$	0	0
$21$	39.7022	0.0196457
$22$	0	0
$23$	−127.036	−0.0500734	−0.0250367	−	0.999687i	$-0.507970\pi$
$23$	−127.036	−0.0500734	−0.0250367	+	0.999687i	$0.507970\pi$
$24$	0	0
$25$	2011.84	0.643789
$26$	0	0
$27$	393.249	0.103815
$28$	0	0
$29$	−2312.40	−0.510584	−0.255292	−	0.966864i	$-0.582172\pi$
$29$	−2312.40	−0.510584	−0.255292	+	0.966864i	$0.582172\pi$
$30$	0	0
$31$	−3992.18	−0.746115	−0.373058	−	0.927808i	$-0.621691\pi$
$31$	−3992.18	−0.746115	−0.373058	+	0.927808i	$0.621691\pi$
$32$	0	0
$33$	461.252	0.0737316
$34$	0	0
$35$	3511.92	0.484589
$36$	0	0
$37$	3853.34	0.462736	0.231368	−	0.972866i	$-0.425680\pi$
$37$	3853.34	0.462736	0.231368	+	0.972866i	$0.425680\pi$
$38$	0	0
$39$	111.573	0.0117462
$40$	0	0
$41$	−4940.82	−0.459029	−0.229514	−	0.973305i	$-0.573714\pi$
$41$	−4940.82	−0.459029	−0.229514	+	0.973305i	$0.573714\pi$
$42$	0	0
$43$	−13679.2	−1.12821	−0.564103	−	0.825704i	$-0.690778\pi$
$43$	−13679.2	−1.12821	−0.564103	+	0.825704i	$0.690778\pi$
$44$	0	0
$45$	17369.2	1.27864
$46$	0	0
$47$	27624.7	1.82412	0.912059	−	0.410058i	$-0.134492\pi$
$47$	27624.7	1.82412	0.912059	+	0.410058i	$0.134492\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	339.216	0.0182621
$52$	0	0
$53$	−37400.0	−1.82886	−0.914432	−	0.404740i	$-0.867362\pi$
$53$	−37400.0	−1.82886	−0.914432	+	0.404740i	$0.867362\pi$
$54$	0	0
$55$	40800.7	1.81870
$56$	0	0
$57$	2067.86	0.0843014
$58$	0	0
$59$	−37000.0	−1.38379	−0.691897	−	0.721996i	$-0.743225\pi$
$59$	−37000.0	−1.38379	−0.691897	+	0.721996i	$0.743225\pi$
$60$	0	0
$61$	3803.32	0.130869	0.0654347	−	0.997857i	$-0.479157\pi$
$61$	3803.32	0.130869	0.0654347	+	0.997857i	$0.479157\pi$
$62$	0	0
$63$	11874.8	0.376943
$64$	0	0
$65$	9869.36	0.289738
$66$	0	0
$67$	22454.7	0.611111	0.305555	−	0.952174i	$-0.401158\pi$
$67$	22454.7	0.611111	0.305555	+	0.952174i	$0.401158\pi$
$68$	0	0
$69$	102.931	0.00260269
$70$	0	0
$71$	55088.9	1.29693	0.648467	−	0.761243i	$-0.275410\pi$
$71$	55088.9	1.29693	0.648467	+	0.761243i	$0.275410\pi$
$72$	0	0
$73$	−69256.6	−1.52109	−0.760543	−	0.649287i	$-0.775067\pi$
$73$	−69256.6	−1.52109	−0.760543	+	0.649287i	$0.775067\pi$
$74$	0	0
$75$	−1630.09	−0.0334625
$76$	0	0
$77$	27894.3	0.536153
$78$	0	0
$79$	40937.8	0.738000	0.369000	−	0.929429i	$-0.379700\pi$
$79$	40937.8	0.738000	0.369000	+	0.929429i	$0.379700\pi$
$80$	0	0
$81$	58570.8	0.991902
$82$	0	0
$83$	19789.0	0.315303	0.157652	−	0.987495i	$-0.449608\pi$
$83$	19789.0	0.315303	0.157652	+	0.987495i	$0.449608\pi$
$84$	0	0
$85$	30005.8	0.450462
$86$	0	0
$87$	1873.62	0.0265389
$88$	0	0
$89$	104148.	1.39373	0.696864	−	0.717203i	$-0.254578\pi$
$89$	104148.	1.39373	0.696864	+	0.717203i	$0.254578\pi$
$90$	0	0
$91$	6747.41	0.0854149
$92$	0	0
$93$	3234.66	0.0387812
$94$	0	0
$95$	182916.	2.07942
$96$	0	0
$97$	−96649.8	−1.04297	−0.521485	−	0.853261i	$-0.674622\pi$
$97$	−96649.8	−1.04297	−0.521485	+	0.853261i	$0.674622\pi$
$98$	0	0
$99$	137959.	1.41469
$100$	0	0
$101$	−26636.2	−0.259817	−0.129909	−	0.991526i	$-0.541468\pi$
$101$	−26636.2	−0.259817	−0.129909	+	0.991526i	$0.541468\pi$
$102$	0	0
$103$	−62232.5	−0.577995	−0.288998	−	0.957330i	$-0.593322\pi$
$103$	−62232.5	−0.577995	−0.288998	+	0.957330i	$0.593322\pi$
$104$	0	0
$105$	−2845.53	−0.0251878
$106$	0	0
$107$	−111445.	−0.941026	−0.470513	−	0.882393i	$-0.655931\pi$
$107$	−111445.	−0.941026	−0.470513	+	0.882393i	$0.655931\pi$
$108$	0	0
$109$	−163349.	−1.31690	−0.658448	−	0.752627i	$-0.728787\pi$
$109$	−163349.	−1.31690	−0.658448	+	0.752627i	$0.728787\pi$
$110$	0	0
$111$	−3122.17	−0.0240519
$112$	0	0
$113$	−180728.	−1.33146	−0.665732	−	0.746191i	$-0.731881\pi$
$113$	−180728.	−1.33146	−0.665732	+	0.746191i	$0.731881\pi$
$114$	0	0
$115$	9104.89	0.0641993
$116$	0	0
$117$	33371.2	0.225376
$118$	0	0
$119$	20514.2	0.132796
$120$	0	0
$121$	163019.	1.01222
$122$	0	0
$123$	4003.30	0.0238592
$124$	0	0
$125$	79782.2	0.456700
$126$	0	0
$127$	121589.	0.668938	0.334469	−	0.942407i	$-0.391443\pi$
$127$	121589.	0.668938	0.334469	+	0.942407i	$0.391443\pi$
$128$	0	0
$129$	11083.5	0.0586414
$130$	0	0
$131$	119712.	0.609482	0.304741	−	0.952435i	$-0.401430\pi$
$131$	119712.	0.609482	0.304741	+	0.952435i	$0.401430\pi$
$132$	0	0
$133$	125054.	0.613014
$134$	0	0
$135$	−28184.9	−0.133101
$136$	0	0
$137$	−4767.80	−0.0217028	−0.0108514	−	0.999941i	$-0.503454\pi$
$137$	−4767.80	−0.0217028	−0.0108514	+	0.999941i	$0.503454\pi$
$138$	0	0
$139$	173481.	0.761578	0.380789	−	0.924662i	$-0.375652\pi$
$139$	173481.	0.761578	0.380789	+	0.924662i	$0.375652\pi$
$140$	0	0
$141$	−22382.9	−0.0948132
$142$	0	0
$143$	78390.0	0.320568
$144$	0	0
$145$	165733.	0.654621
$146$	0	0
$147$	−1945.41	−0.00742536
$148$	0	0
$149$	236401.	0.872337	0.436169	−	0.899865i	$-0.356335\pi$
$149$	236401.	0.872337	0.436169	+	0.899865i	$0.356335\pi$
$150$	0	0
$151$	−280973.	−1.00282	−0.501410	−	0.865210i	$-0.667185\pi$
$151$	−280973.	−1.00282	−0.501410	+	0.865210i	$0.667185\pi$
$152$	0	0
$153$	101459.	0.350397
$154$	0	0
$155$	286127.	0.956597
$156$	0	0
$157$	−343000.	−1.11057	−0.555284	−	0.831661i	$-0.687390\pi$
$157$	−343000.	−1.11057	−0.555284	+	0.831661i	$0.687390\pi$
$158$	0	0
$159$	30303.3	0.0950598
$160$	0	0
$161$	6224.76	0.0189260
$162$	0	0
$163$	631306.	1.86110	0.930552	−	0.366160i	$-0.119328\pi$
$163$	631306.	1.86110	0.930552	+	0.366160i	$0.119328\pi$
$164$	0	0
$165$	−33058.7	−0.0945315
$166$	0	0
$167$	174139.	0.483176	0.241588	−	0.970379i	$-0.422332\pi$
$167$	174139.	0.483176	0.241588	+	0.970379i	$0.422332\pi$
$168$	0	0
$169$	−352331.	−0.948930
$170$	0	0
$171$	618493.	1.61750
$172$	0	0
$173$	389487.	0.989412	0.494706	−	0.869060i	$-0.335276\pi$
$173$	389487.	0.989412	0.494706	+	0.869060i	$0.335276\pi$
$174$	0	0
$175$	−98580.1	−0.243329
$176$	0	0
$177$	29979.2	0.0719262
$178$	0	0
$179$	827314.	1.92991	0.964956	−	0.262411i	$-0.0845178\pi$
$179$	827314.	1.92991	0.964956	+	0.262411i	$0.0845178\pi$
$180$	0	0
$181$	−250850.	−0.569138	−0.284569	−	0.958656i	$-0.591850\pi$
$181$	−250850.	−0.569138	−0.284569	+	0.958656i	$0.591850\pi$
$182$	0	0
$183$	−3081.64	−0.00680227
$184$	0	0
$185$	−276176.	−0.593275
$186$	0	0
$187$	238329.	0.498395
$188$	0	0
$189$	−19269.2	−0.0392383
$190$	0	0
$191$	166405.	0.330052	0.165026	−	0.986289i	$-0.447229\pi$
$191$	166405.	0.330052	0.165026	+	0.986289i	$0.447229\pi$
$192$	0	0
$193$	−408270.	−0.788959	−0.394480	−	0.918905i	$-0.629075\pi$
$193$	−408270.	−0.788959	−0.394480	+	0.918905i	$0.629075\pi$
$194$	0	0
$195$	−7996.65	−0.0150599
$196$	0	0
$197$	−895496.	−1.64399	−0.821993	−	0.569497i	$-0.807138\pi$
$197$	−895496.	−1.64399	−0.821993	+	0.569497i	$0.807138\pi$
$198$	0	0
$199$	177488.	0.317714	0.158857	−	0.987302i	$-0.449219\pi$
$199$	177488.	0.317714	0.158857	+	0.987302i	$0.449219\pi$
$200$	0	0
$201$	−18193.9	−0.0317640
$202$	0	0
$203$	113307.	0.192983
$204$	0	0
$205$	354117.	0.588522
$206$	0	0
$207$	30786.3	0.0499381
$208$	0	0
$209$	1.45286e6	2.30068
$210$	0	0
$211$	−963464.	−1.48981	−0.744903	−	0.667173i	$-0.767504\pi$
$211$	−963464.	−1.48981	−0.744903	+	0.667173i	$0.767504\pi$
$212$	0	0
$213$	−44635.7	−0.0674114
$214$	0	0
$215$	980410.	1.44648
$216$	0	0
$217$	195617.	0.282005
$218$	0	0
$219$	56115.1	0.0790623
$220$	0	0
$221$	57649.9	0.0793996
$222$	0	0
$223$	−513529.	−0.691518	−0.345759	−	0.938323i	$-0.612378\pi$
$223$	−513529.	−0.691518	−0.345759	+	0.938323i	$0.612378\pi$
$224$	0	0
$225$	−487556.	−0.642049
$226$	0	0
$227$	251707.	0.324213	0.162106	−	0.986773i	$-0.448171\pi$
$227$	251707.	0.324213	0.162106	+	0.986773i	$0.448171\pi$
$228$	0	0
$229$	1.11802e6	1.40884	0.704419	−	0.709784i	$-0.251207\pi$
$229$	1.11802e6	1.40884	0.704419	+	0.709784i	$0.251207\pi$
$230$	0	0
$231$	−22601.3	−0.0278679
$232$	0	0
$233$	−1.09444e6	−1.32070	−0.660349	−	0.750959i	$-0.729592\pi$
$233$	−1.09444e6	−1.32070	−0.660349	+	0.750959i	$0.729592\pi$
$234$	0	0
$235$	−1.97991e6	−2.33871
$236$	0	0
$237$	−33169.8	−0.0383594
$238$	0	0
$239$	−1.61769e6	−1.83190	−0.915949	−	0.401295i	$-0.868560\pi$
$239$	−1.61769e6	−1.83190	−0.915949	+	0.401295i	$0.868560\pi$
$240$	0	0
$241$	368001.	0.408137	0.204069	−	0.978957i	$-0.434583\pi$
$241$	368001.	0.408137	0.204069	+	0.978957i	$0.434583\pi$
$242$	0	0
$243$	−143017.	−0.155371
$244$	0	0
$245$	−172084.	−0.183158
$246$	0	0
$247$	351434.	0.366523
$248$	0	0
$249$	−16034.0	−0.0163887
$250$	0	0
$251$	132644.	0.132893	0.0664466	−	0.997790i	$-0.478834\pi$
$251$	132644.	0.132893	0.0664466	+	0.997790i	$0.478834\pi$
$252$	0	0
$253$	72318.0	0.0710305
$254$	0	0
$255$	−24312.2	−0.0234139
$256$	0	0
$257$	−648941.	−0.612876	−0.306438	−	0.951891i	$-0.599137\pi$
$257$	−648941.	−0.612876	−0.306438	+	0.951891i	$0.599137\pi$
$258$	0	0
$259$	−188814.	−0.174898
$260$	0	0
$261$	560394.	0.509205
$262$	0	0
$263$	−559540.	−0.498818	−0.249409	−	0.968398i	$-0.580236\pi$
$263$	−559540.	−0.498818	−0.249409	+	0.968398i	$0.580236\pi$
$264$	0	0
$265$	2.68052e6	2.34479
$266$	0	0
$267$	−84386.3	−0.0724425
$268$	0	0
$269$	−1.11891e6	−0.942793	−0.471397	−	0.881921i	$-0.656250\pi$
$269$	−1.11891e6	−0.942793	−0.471397	+	0.881921i	$0.656250\pi$
$270$	0	0
$271$	−1.85235e6	−1.53214	−0.766072	−	0.642754i	$-0.777792\pi$
$271$	−1.85235e6	−1.53214	−0.766072	+	0.642754i	$0.777792\pi$
$272$	0	0
$273$	−5467.09	−0.00443966
$274$	0	0
$275$	−1.14528e6	−0.913232
$276$	0	0
$277$	150989.	0.118235	0.0591173	−	0.998251i	$-0.481171\pi$
$277$	150989.	0.118235	0.0591173	+	0.998251i	$0.481171\pi$
$278$	0	0
$279$	967479.	0.744099
$280$	0	0
$281$	−658293.	−0.497340	−0.248670	−	0.968588i	$-0.579994\pi$
$281$	−658293.	−0.497340	−0.248670	+	0.968588i	$0.579994\pi$
$282$	0	0
$283$	−1.88185e6	−1.39675	−0.698376	−	0.715731i	$-0.746094\pi$
$283$	−1.88185e6	−1.39675	−0.698376	+	0.715731i	$0.746094\pi$
$284$	0	0
$285$	−148207.	−0.108083
$286$	0	0
$287$	242100.	0.173496
$288$	0	0
$289$	−1.24458e6	−0.876556
$290$	0	0
$291$	78310.5	0.0542110
$292$	0	0
$293$	−1.04860e6	−0.713577	−0.356789	−	0.934185i	$-0.616128\pi$
$293$	−1.04860e6	−0.713577	−0.356789	+	0.934185i	$0.616128\pi$
$294$	0	0
$295$	2.65185e6	1.77417
$296$	0	0
$297$	−223866.	−0.147264
$298$	0	0
$299$	17493.1	0.0113159
$300$	0	0
$301$	670279.	0.426422
$302$	0	0
$303$	21581.9	0.0135047
$304$	0	0
$305$	−272591.	−0.167788
$306$	0	0
$307$	−2.11586e6	−1.28127	−0.640634	−	0.767846i	$-0.721329\pi$
$307$	−2.11586e6	−1.28127	−0.640634	+	0.767846i	$0.721329\pi$
$308$	0	0
$309$	50423.9	0.0300428
$310$	0	0
$311$	1.58928e6	0.931749	0.465875	−	0.884851i	$-0.345740\pi$
$311$	1.58928e6	0.931749	0.465875	+	0.884851i	$0.345740\pi$
$312$	0	0
$313$	150833.	0.0870235	0.0435118	−	0.999053i	$-0.486145\pi$
$313$	150833.	0.0870235	0.0435118	+	0.999053i	$0.486145\pi$
$314$	0	0
$315$	−851090.	−0.483280
$316$	0	0
$317$	17178.1	0.00960122	0.00480061	−	0.999988i	$-0.498472\pi$
$317$	17178.1	0.00960122	0.00480061	+	0.999988i	$0.498472\pi$
$318$	0	0
$319$	1.31638e6	0.724277
$320$	0	0
$321$	90298.4	0.0489122
$322$	0	0
$323$	1.06847e6	0.569842
$324$	0	0
$325$	−277035.	−0.145488
$326$	0	0
$327$	132354.	0.0684490
$328$	0	0
$329$	−1.35361e6	−0.689452
$330$	0	0
$331$	−2.47433e6	−1.24133	−0.620666	−	0.784075i	$-0.713138\pi$
$331$	−2.47433e6	−1.24133	−0.620666	+	0.784075i	$0.713138\pi$
$332$	0	0
$333$	−933833.	−0.461486
$334$	0	0
$335$	−1.60937e6	−0.783507
$336$	0	0
$337$	−1.92911e6	−0.925302	−0.462651	−	0.886541i	$-0.653102\pi$
$337$	−1.92911e6	−0.925302	−0.462651	+	0.886541i	$0.653102\pi$
$338$	0	0
$339$	146435.	0.0692062
$340$	0	0
$341$	2.27263e6	1.05838
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−7377.24	−0.00333692
$346$	0	0
$347$	566575.	0.252600	0.126300	−	0.991992i	$-0.459690\pi$
$347$	566575.	0.252600	0.126300	+	0.991992i	$0.459690\pi$
$348$	0	0
$349$	−1.91470e6	−0.841469	−0.420734	−	0.907184i	$-0.638228\pi$
$349$	−1.91470e6	−0.841469	−0.420734	+	0.907184i	$0.638228\pi$
$350$	0	0
$351$	−54151.3	−0.0234607
$352$	0	0
$353$	3.75846e6	1.60536	0.802681	−	0.596409i	$-0.203406\pi$
$353$	3.75846e6	1.60536	0.802681	+	0.596409i	$0.203406\pi$
$354$	0	0
$355$	−3.94831e6	−1.66280
$356$	0	0
$357$	−16621.6	−0.00690243
$358$	0	0
$359$	−2.88170e6	−1.18008	−0.590042	−	0.807373i	$-0.700889\pi$
$359$	−2.88170e6	−1.18008	−0.590042	+	0.807373i	$0.700889\pi$
$360$	0	0
$361$	4.03728e6	1.63050
$362$	0	0
$363$	−132086.	−0.0526127
$364$	0	0
$365$	4.96374e6	1.95019
$366$	0	0
$367$	−4.00494e6	−1.55214	−0.776070	−	0.630647i	$-0.782789\pi$
$367$	−4.00494e6	−1.55214	−0.776070	+	0.630647i	$0.782789\pi$
$368$	0	0
$369$	1.19738e6	0.457788
$370$	0	0
$371$	1.83260e6	0.691246
$372$	0	0
$373$	527856.	0.196446	0.0982230	−	0.995164i	$-0.468684\pi$
$373$	527856.	0.196446	0.0982230	+	0.995164i	$0.468684\pi$
$374$	0	0
$375$	−64643.5	−0.0237381
$376$	0	0
$377$	318422.	0.115385
$378$	0	0
$379$	−771013.	−0.275717	−0.137858	−	0.990452i	$-0.544022\pi$
$379$	−771013.	−0.275717	−0.137858	+	0.990452i	$0.544022\pi$
$380$	0	0
$381$	−98517.6	−0.0347697
$382$	0	0
$383$	−1.49145e6	−0.519530	−0.259765	−	0.965672i	$-0.583645\pi$
$383$	−1.49145e6	−0.519530	−0.259765	+	0.965672i	$0.583645\pi$
$384$	0	0
$385$	−1.99923e6	−0.687403
$386$	0	0
$387$	3.31506e6	1.12516
$388$	0	0
$389$	−4.84613e6	−1.62376	−0.811878	−	0.583827i	$-0.801555\pi$
$389$	−4.84613e6	−1.62376	−0.811878	+	0.583827i	$0.801555\pi$
$390$	0	0
$391$	53184.4	0.0175931
$392$	0	0
$393$	−96997.0	−0.0316794
$394$	0	0
$395$	−2.93408e6	−0.946192
$396$	0	0
$397$	−241537.	−0.0769144	−0.0384572	−	0.999260i	$-0.512244\pi$
$397$	−241537.	−0.0769144	−0.0384572	+	0.999260i	$0.512244\pi$
$398$	0	0
$399$	−101325.	−0.0318629
$400$	0	0
$401$	−3.33211e6	−1.03481	−0.517403	−	0.855742i	$-0.673101\pi$
$401$	−3.33211e6	−1.03481	−0.517403	+	0.855742i	$0.673101\pi$
$402$	0	0
$403$	549732.	0.168612
$404$	0	0
$405$	−4.19787e6	−1.27172
$406$	0	0
$407$	−2.19360e6	−0.656404
$408$	0	0
$409$	−4.42388e6	−1.30766	−0.653831	−	0.756641i	$-0.726839\pi$
$409$	−4.42388e6	−1.30766	−0.653831	+	0.756641i	$0.726839\pi$
$410$	0	0
$411$	3863.11	0.00112806
$412$	0	0
$413$	1.81300e6	0.523025
$414$	0	0
$415$	−1.41831e6	−0.404251
$416$	0	0
$417$	−140563.	−0.0395849
$418$	0	0
$419$	5.05733e6	1.40730	0.703649	−	0.710548i	$-0.251553\pi$
$419$	5.05733e6	1.40730	0.703649	+	0.710548i	$0.251553\pi$
$420$	0	0
$421$	6.46515e6	1.77776	0.888881	−	0.458139i	$-0.151484\pi$
$421$	6.46515e6	1.77776	0.888881	+	0.458139i	$0.151484\pi$
$422$	0	0
$423$	−6.69467e6	−1.81919
$424$	0	0
$425$	−842270.	−0.226193
$426$	0	0
$427$	−186363.	−0.0494640
$428$	0	0
$429$	−63515.4	−0.0166623
$430$	0	0
$431$	−4.92475e6	−1.27700	−0.638500	−	0.769622i	$-0.720445\pi$
$431$	−4.92475e6	−1.27700	−0.638500	+	0.769622i	$0.720445\pi$
$432$	0	0
$433$	2.99870e6	0.768622	0.384311	−	0.923204i	$-0.374439\pi$
$433$	2.99870e6	0.768622	0.384311	+	0.923204i	$0.374439\pi$
$434$	0	0
$435$	−134286.	−0.0340256
$436$	0	0
$437$	324213.	0.0812131
$438$	0	0
$439$	5.09504e6	1.26179	0.630894	−	0.775869i	$-0.282688\pi$
$439$	5.09504e6	1.26179	0.630894	+	0.775869i	$0.282688\pi$
$440$	0	0
$441$	−581867.	−0.142471
$442$	0	0
$443$	−2.63893e6	−0.638878	−0.319439	−	0.947607i	$-0.603494\pi$
$443$	−2.63893e6	−0.638878	−0.319439	+	0.947607i	$0.603494\pi$
$444$	0	0
$445$	−7.46450e6	−1.78690
$446$	0	0
$447$	−191544.	−0.0453419
$448$	0	0
$449$	5.01164e6	1.17318	0.586589	−	0.809885i	$-0.300470\pi$
$449$	5.01164e6	1.17318	0.586589	+	0.809885i	$0.300470\pi$
$450$	0	0
$451$	2.81267e6	0.651145
$452$	0	0
$453$	227659.	0.0521241
$454$	0	0
$455$	−483599.	−0.109511
$456$	0	0
$457$	−5.62960e6	−1.26092	−0.630460	−	0.776222i	$-0.717134\pi$
$457$	−5.62960e6	−1.26092	−0.630460	+	0.776222i	$0.717134\pi$
$458$	0	0
$459$	−164636.	−0.0364749
$460$	0	0
$461$	−4.21775e6	−0.924334	−0.462167	−	0.886793i	$-0.652928\pi$
$461$	−4.21775e6	−0.924334	−0.462167	+	0.886793i	$0.652928\pi$
$462$	0	0
$463$	−1.79250e6	−0.388603	−0.194301	−	0.980942i	$-0.562244\pi$
$463$	−1.79250e6	−0.388603	−0.194301	+	0.980942i	$0.562244\pi$
$464$	0	0
$465$	−231834.	−0.0497215
$466$	0	0
$467$	5.56530e6	1.18085	0.590427	−	0.807091i	$-0.298959\pi$
$467$	5.56530e6	1.18085	0.590427	+	0.807091i	$0.298959\pi$
$468$	0	0
$469$	−1.10028e6	−0.230978
$470$	0	0
$471$	277916.	0.0577246
$472$	0	0
$473$	7.78716e6	1.60039
$474$	0	0
$475$	−5.13448e6	−1.04415
$476$	0	0
$477$	9.06363e6	1.82392
$478$	0	0
$479$	7.98318e6	1.58978	0.794890	−	0.606753i	$-0.207528\pi$
$479$	7.98318e6	1.58978	0.794890	+	0.606753i	$0.207528\pi$
$480$	0	0
$481$	−530614.	−0.104572
$482$	0	0
$483$	−5043.61	−0.000983726 0
$484$	0	0
$485$	6.92706e6	1.33719
$486$	0	0
$487$	1.89273e6	0.361631	0.180815	−	0.983517i	$-0.442126\pi$
$487$	1.89273e6	0.361631	0.180815	+	0.983517i	$0.442126\pi$
$488$	0	0
$489$	−511515.	−0.0967356
$490$	0	0
$491$	−714559.	−0.133762	−0.0668812	−	0.997761i	$-0.521305\pi$
$491$	−714559.	−0.133762	−0.0668812	+	0.997761i	$0.521305\pi$
$492$	0	0
$493$	968100.	0.179392
$494$	0	0
$495$	−9.88778e6	−1.81378
$496$	0	0
$497$	−2.69935e6	−0.490195
$498$	0	0
$499$	6.55243e6	1.17802	0.589008	−	0.808127i	$-0.299519\pi$
$499$	6.55243e6	1.17802	0.589008	+	0.808127i	$0.299519\pi$
$500$	0	0
$501$	−141096.	−0.0251143
$502$	0	0
$503$	990048.	0.174476	0.0872381	−	0.996187i	$-0.472196\pi$
$503$	990048.	0.174476	0.0872381	+	0.996187i	$0.472196\pi$
$504$	0	0
$505$	1.90906e6	0.333113
$506$	0	0
$507$	285476.	0.0493231
$508$	0	0
$509$	5.01675e6	0.858277	0.429139	−	0.903239i	$-0.358817\pi$
$509$	5.01675e6	0.858277	0.429139	+	0.903239i	$0.358817\pi$
$510$	0	0
$511$	3.39357e6	0.574917
$512$	0	0
$513$	−1.00362e6	−0.168375
$514$	0	0
$515$	4.46031e6	0.741049
$516$	0	0
$517$	−1.57260e7	−2.58756
$518$	0	0
$519$	−315581.	−0.0514272
$520$	0	0
$521$	−196789.	−0.0317619	−0.0158809	−	0.999874i	$-0.505055\pi$
$521$	−196789.	−0.0317619	−0.0158809	+	0.999874i	$0.505055\pi$
$522$	0	0
$523$	4.81508e6	0.769749	0.384875	−	0.922969i	$-0.374245\pi$
$523$	4.81508e6	0.769749	0.384875	+	0.922969i	$0.374245\pi$
$524$	0	0
$525$	79874.5	0.0126477
$526$	0	0
$527$	1.67135e6	0.262145
$528$	0	0
$529$	−6.42020e6	−0.997493
$530$	0	0
$531$	8.96671e6	1.38006
$532$	0	0
$533$	680362.	0.103734
$534$	0	0
$535$	7.98747e6	1.20649
$536$	0	0
$537$	−670330.	−0.100312
$538$	0	0
$539$	−1.36682e6	−0.202647
$540$	0	0
$541$	−2.47799e6	−0.364005	−0.182002	−	0.983298i	$-0.558258\pi$
$541$	−2.47799e6	−0.364005	−0.182002	+	0.983298i	$0.558258\pi$
$542$	0	0
$543$	203251.	0.0295824
$544$	0	0
$545$	1.17075e7	1.68840
$546$	0	0
$547$	5.97032e6	0.853158	0.426579	−	0.904450i	$-0.359719\pi$
$547$	5.97032e6	0.853158	0.426579	+	0.904450i	$0.359719\pi$
$548$	0	0
$549$	−921710.	−0.130516
$550$	0	0
$551$	5.90154e6	0.828107
$552$	0	0
$553$	−2.00595e6	−0.278938
$554$	0	0
$555$	223771.	0.0308370
$556$	0	0
$557$	−7.59368e6	−1.03708	−0.518542	−	0.855052i	$-0.673525\pi$
$557$	−7.59368e6	−1.03708	−0.518542	+	0.855052i	$0.673525\pi$
$558$	0	0
$559$	1.88365e6	0.254960
$560$	0	0
$561$	−193106.	−0.0259053
$562$	0	0
$563$	1.11744e7	1.48578	0.742888	−	0.669416i	$-0.233456\pi$
$563$	1.11744e7	1.48578	0.742888	+	0.669416i	$0.233456\pi$
$564$	0	0
$565$	1.29531e7	1.70707
$566$	0	0
$567$	−2.86997e6	−0.374904
$568$	0	0
$569$	−1.18434e7	−1.53354	−0.766771	−	0.641921i	$-0.778138\pi$
$569$	−1.18434e7	−1.53354	−0.766771	+	0.641921i	$0.778138\pi$
$570$	0	0
$571$	−1.00356e7	−1.28811	−0.644057	−	0.764977i	$-0.722750\pi$
$571$	−1.00356e7	−1.28811	−0.644057	+	0.764977i	$0.722750\pi$
$572$	0	0
$573$	−134829.	−0.0171553
$574$	0	0
$575$	−255576.	−0.0322367
$576$	0	0
$577$	1.12162e7	1.40251	0.701256	−	0.712909i	$-0.252623\pi$
$577$	1.12162e7	1.40251	0.701256	+	0.712909i	$0.252623\pi$
$578$	0	0
$579$	330801.	0.0410082
$580$	0	0
$581$	−969661.	−0.119173
$582$	0	0
$583$	2.12907e7	2.59429
$584$	0	0
$585$	−2.39178e6	−0.288955
$586$	0	0
$587$	−3.16178e6	−0.378736	−0.189368	−	0.981906i	$-0.560644\pi$
$587$	−3.16178e6	−0.378736	−0.189368	+	0.981906i	$0.560644\pi$
$588$	0	0
$589$	1.01886e7	1.21011
$590$	0	0
$591$	725575.	0.0854504
$592$	0	0
$593$	−7.55501e6	−0.882264	−0.441132	−	0.897442i	$-0.645423\pi$
$593$	−7.55501e6	−0.882264	−0.441132	+	0.897442i	$0.645423\pi$
$594$	0	0
$595$	−1.47029e6	−0.170259
$596$	0	0
$597$	−143810.	−0.0165140
$598$	0	0
$599$	1.56629e7	1.78364	0.891818	−	0.452394i	$-0.149430\pi$
$599$	1.56629e7	1.78364	0.891818	+	0.452394i	$0.149430\pi$
$600$	0	0
$601$	1.07168e7	1.21027	0.605133	−	0.796125i	$-0.293120\pi$
$601$	1.07168e7	1.21027	0.605133	+	0.796125i	$0.293120\pi$
$602$	0	0
$603$	−5.44175e6	−0.609460
$604$	0	0
$605$	−1.16839e7	−1.29777
$606$	0	0
$607$	1.24101e7	1.36711	0.683557	−	0.729897i	$-0.260432\pi$
$607$	1.24101e7	1.36711	0.683557	+	0.729897i	$0.260432\pi$
$608$	0	0
$609$	−91807.3	−0.0100308
$610$	0	0
$611$	−3.80398e6	−0.412226
$612$	0	0
$613$	8.92284e6	0.959074	0.479537	−	0.877522i	$-0.340805\pi$
$613$	8.92284e6	0.959074	0.479537	+	0.877522i	$0.340805\pi$
$614$	0	0
$615$	−286924.	−0.0305899
$616$	0	0
$617$	5.46678e6	0.578121	0.289060	−	0.957311i	$-0.406657\pi$
$617$	5.46678e6	0.578121	0.289060	+	0.957311i	$0.406657\pi$
$618$	0	0
$619$	−4.14856e6	−0.435181	−0.217591	−	0.976040i	$-0.569820\pi$
$619$	−4.14856e6	−0.435181	−0.217591	+	0.976040i	$0.569820\pi$
$620$	0	0
$621$	−49956.8	−0.00519835
$622$	0	0
$623$	−5.10328e6	−0.526780
$624$	0	0
$625$	−1.20051e7	−1.22932
$626$	0	0
$627$	−1.17718e6	−0.119584
$628$	0	0
$629$	−1.61323e6	−0.162581
$630$	0	0
$631$	1.66340e7	1.66312	0.831561	−	0.555433i	$-0.187447\pi$
$631$	1.66340e7	1.66312	0.831561	+	0.555433i	$0.187447\pi$
$632$	0	0
$633$	780647.	0.0774364
$634$	0	0
$635$	−8.71451e6	−0.857647
$636$	0	0
$637$	−330623.	−0.0322838
$638$	0	0
$639$	−1.33504e7	−1.29343
$640$	0	0
$641$	−1.54577e7	−1.48594	−0.742968	−	0.669327i	$-0.766582\pi$
$641$	−1.54577e7	−1.48594	−0.742968	+	0.669327i	$0.766582\pi$
$642$	0	0
$643$	−1.42448e7	−1.35872	−0.679360	−	0.733805i	$-0.737743\pi$
$643$	−1.42448e7	−1.35872	−0.679360	+	0.733805i	$0.737743\pi$
$644$	0	0
$645$	−794377.	−0.0751843
$646$	0	0
$647$	9.78031e6	0.918527	0.459264	−	0.888300i	$-0.348113\pi$
$647$	9.78031e6	0.918527	0.459264	+	0.888300i	$0.348113\pi$
$648$	0	0
$649$	2.10630e7	1.96295
$650$	0	0
$651$	−158498.	−0.0146579
$652$	0	0
$653$	−2.43600e6	−0.223560	−0.111780	−	0.993733i	$-0.535655\pi$
$653$	−2.43600e6	−0.223560	−0.111780	+	0.993733i	$0.535655\pi$
$654$	0	0
$655$	−8.58000e6	−0.781419
$656$	0	0
$657$	1.67839e7	1.51698
$658$	0	0
$659$	3.49022e6	0.313069	0.156534	−	0.987673i	$-0.449968\pi$
$659$	3.49022e6	0.313069	0.156534	+	0.987673i	$0.449968\pi$
$660$	0	0
$661$	−1.38495e7	−1.23291	−0.616453	−	0.787392i	$-0.711431\pi$
$661$	−1.38495e7	−1.23291	−0.616453	+	0.787392i	$0.711431\pi$
$662$	0	0
$663$	−46710.8	−0.00412699
$664$	0	0
$665$	−8.96287e6	−0.785947
$666$	0	0
$667$	293758.	0.0255667
$668$	0	0
$669$	416087.	0.0359434
$670$	0	0
$671$	−2.16512e6	−0.185642
$672$	0	0
$673$	−1.47132e7	−1.25219	−0.626093	−	0.779748i	$-0.715347\pi$
$673$	−1.47132e7	−1.25219	−0.626093	+	0.779748i	$0.715347\pi$
$674$	0	0
$675$	791155.	0.0668347
$676$	0	0
$677$	2.00252e7	1.67921	0.839605	−	0.543198i	$-0.182787\pi$
$677$	2.00252e7	1.67921	0.839605	+	0.543198i	$0.182787\pi$
$678$	0	0
$679$	4.73584e6	0.394205
$680$	0	0
$681$	−203945.	−0.0168518
$682$	0	0
$683$	1.84454e7	1.51299	0.756495	−	0.654000i	$-0.226910\pi$
$683$	1.84454e7	1.51299	0.756495	+	0.654000i	$0.226910\pi$
$684$	0	0
$685$	341717.	0.0278253
$686$	0	0
$687$	−905876.	−0.0732280
$688$	0	0
$689$	5.15006e6	0.413299
$690$	0	0
$691$	−5.52332e6	−0.440053	−0.220027	−	0.975494i	$-0.570614\pi$
$691$	−5.52332e6	−0.440053	−0.220027	+	0.975494i	$0.570614\pi$
$692$	0	0
$693$	−6.76000e6	−0.534704
$694$	0	0
$695$	−1.24337e7	−0.976421
$696$	0	0
$697$	2.06851e6	0.161278
$698$	0	0
$699$	886772.	0.0686466
$700$	0	0
$701$	−5.11961e6	−0.393497	−0.196749	−	0.980454i	$-0.563038\pi$
$701$	−5.11961e6	−0.393497	−0.196749	+	0.980454i	$0.563038\pi$
$702$	0	0
$703$	−9.83424e6	−0.750503
$704$	0	0
$705$	1.60422e6	0.121560
$706$	0	0
$707$	1.30517e6	0.0982017
$708$	0	0
$709$	−2.98577e6	−0.223070	−0.111535	−	0.993761i	$-0.535577\pi$
$709$	−2.98577e6	−0.223070	−0.111535	+	0.993761i	$0.535577\pi$
$710$	0	0
$711$	−9.92100e6	−0.736006
$712$	0	0
$713$	507151.	0.0373605
$714$	0	0
$715$	−5.61835e6	−0.411001
$716$	0	0
$717$	1.31073e6	0.0952175
$718$	0	0
$719$	−2.51554e6	−0.181472	−0.0907360	−	0.995875i	$-0.528922\pi$
$719$	−2.51554e6	−0.181472	−0.0907360	+	0.995875i	$0.528922\pi$
$720$	0	0
$721$	3.04939e6	0.218462
$722$	0	0
$723$	−298173.	−0.0212140
$724$	0	0
$725$	−4.65217e6	−0.328708
$726$	0	0
$727$	1.55008e7	1.08772	0.543862	−	0.839175i	$-0.316962\pi$
$727$	1.55008e7	1.08772	0.543862	+	0.839175i	$0.316962\pi$
$728$	0	0
$729$	−1.41168e7	−0.983826
$730$	0	0
$731$	5.72687e6	0.396391
$732$	0	0
$733$	1.88611e7	1.29660	0.648302	−	0.761384i	$-0.275480\pi$
$733$	1.88611e7	1.29660	0.648302	+	0.761384i	$0.275480\pi$
$734$	0	0
$735$	139431.	0.00952008
$736$	0	0
$737$	−1.27828e7	−0.866877
$738$	0	0
$739$	−2.73270e6	−0.184069	−0.0920346	−	0.995756i	$-0.529337\pi$
$739$	−2.73270e6	−0.184069	−0.0920346	+	0.995756i	$0.529337\pi$
$740$	0	0
$741$	−284749.	−0.0190510
$742$	0	0
$743$	108079.	0.00718239	0.00359119	−	0.999994i	$-0.498857\pi$
$743$	108079.	0.00718239	0.00359119	+	0.999994i	$0.498857\pi$
$744$	0	0
$745$	−1.69433e7	−1.11843
$746$	0	0
$747$	−4.79574e6	−0.314452
$748$	0	0
$749$	5.46081e6	0.355675
$750$	0	0
$751$	1.41662e7	0.916543	0.458272	−	0.888812i	$-0.348469\pi$
$751$	1.41662e7	0.916543	0.458272	+	0.888812i	$0.348469\pi$
$752$	0	0
$753$	−107475.	−0.00690747
$754$	0	0
$755$	2.01379e7	1.28572
$756$	0	0
$757$	1.35016e6	0.0856339	0.0428169	−	0.999083i	$-0.486367\pi$
$757$	1.35016e6	0.0856339	0.0428169	+	0.999083i	$0.486367\pi$
$758$	0	0
$759$	−58595.6	−0.00369199
$760$	0	0
$761$	1.16646e7	0.730146	0.365073	−	0.930979i	$-0.381044\pi$
$761$	1.16646e7	0.730146	0.365073	+	0.930979i	$0.381044\pi$
$762$	0	0
$763$	8.00412e6	0.497740
$764$	0	0
$765$	−7.27172e6	−0.449245
$766$	0	0
$767$	5.09498e6	0.312719
$768$	0	0
$769$	1.99473e7	1.21638	0.608188	−	0.793793i	$-0.291897\pi$
$769$	1.99473e7	1.21638	0.608188	+	0.793793i	$0.291897\pi$
$770$	0	0
$771$	525804.	0.0318558
$772$	0	0
$773$	−1.52976e7	−0.920818	−0.460409	−	0.887707i	$-0.652297\pi$
$773$	−1.52976e7	−0.920818	−0.460409	+	0.887707i	$0.652297\pi$
$774$	0	0
$775$	−8.03163e6	−0.480340
$776$	0	0
$777$	152986.	0.00909076
$778$	0	0
$779$	1.26096e7	0.744490
$780$	0	0
$781$	−3.13605e7	−1.83974
$782$	0	0
$783$	−909348.	−0.0530061
$784$	0	0
$785$	2.45834e7	1.42386
$786$	0	0
$787$	−3.34094e7	−1.92279	−0.961394	−	0.275176i	$-0.911264\pi$
$787$	−3.34094e7	−1.92279	−0.961394	+	0.275176i	$0.911264\pi$
$788$	0	0
$789$	453367.	0.0259273
$790$	0	0
$791$	8.85568e6	0.503246
$792$	0	0
$793$	−523726.	−0.0295747
$794$	0	0
$795$	−2.17189e6	−0.121877
$796$	0	0
$797$	3.12154e7	1.74070	0.870349	−	0.492436i	$-0.163893\pi$
$797$	3.12154e7	1.74070	0.870349	+	0.492436i	$0.163893\pi$
$798$	0	0
$799$	−1.15653e7	−0.640898
$800$	0	0
$801$	−2.52397e7	−1.38996
$802$	0	0
$803$	3.94258e7	2.15770
$804$	0	0
$805$	−446140.	−0.0242651
$806$	0	0
$807$	906600.	0.0490041
$808$	0	0
$809$	7.90944e6	0.424888	0.212444	−	0.977173i	$-0.431858\pi$
$809$	7.90944e6	0.424888	0.212444	+	0.977173i	$0.431858\pi$
$810$	0	0
$811$	4.94124e6	0.263805	0.131903	−	0.991263i	$-0.457891\pi$
$811$	4.94124e6	0.263805	0.131903	+	0.991263i	$0.457891\pi$
$812$	0	0
$813$	1.50087e6	0.0796371
$814$	0	0
$815$	−4.52468e7	−2.38613
$816$	0	0
$817$	3.49110e7	1.82982
$818$	0	0
$819$	−1.63519e6	−0.0851841
$820$	0	0
$821$	−4.08881e6	−0.211709	−0.105855	−	0.994382i	$-0.533758\pi$
$821$	−4.08881e6	−0.211709	−0.105855	+	0.994382i	$0.533758\pi$
$822$	0	0
$823$	9.11809e6	0.469250	0.234625	−	0.972086i	$-0.424614\pi$
$823$	9.11809e6	0.469250	0.234625	+	0.972086i	$0.424614\pi$
$824$	0	0
$825$	927965.	0.0474675
$826$	0	0
$827$	8.21913e6	0.417890	0.208945	−	0.977927i	$-0.432997\pi$
$827$	8.21913e6	0.417890	0.208945	+	0.977927i	$0.432997\pi$
$828$	0	0
$829$	2.40529e7	1.21557	0.607787	−	0.794100i	$-0.292057\pi$
$829$	2.40529e7	1.21557	0.607787	+	0.794100i	$0.292057\pi$
$830$	0	0
$831$	−122338.	−0.00614555
$832$	0	0
$833$	−1.00519e6	−0.0501923
$834$	0	0
$835$	−1.24809e7	−0.619482
$836$	0	0
$837$	−1.56992e6	−0.0774577
$838$	0	0
$839$	−4.55100e6	−0.223204	−0.111602	−	0.993753i	$-0.535598\pi$
$839$	−4.55100e6	−0.223204	−0.111602	+	0.993753i	$0.535598\pi$
$840$	0	0
$841$	−1.51640e7	−0.739304
$842$	0	0
$843$	533382.	0.0258505
$844$	0	0
$845$	2.52522e7	1.21663
$846$	0	0
$847$	−7.98793e6	−0.382583
$848$	0	0
$849$	1.52477e6	0.0725998
$850$	0	0
$851$	−489513.	−0.0231708
$852$	0	0
$853$	1.92326e7	0.905033	0.452517	−	0.891756i	$-0.350526\pi$
$853$	1.92326e7	0.905033	0.452517	+	0.891756i	$0.350526\pi$
$854$	0	0
$855$	−4.43284e7	−2.07380
$856$	0	0
$857$	−2.44005e7	−1.13487	−0.567435	−	0.823418i	$-0.692064\pi$
$857$	−2.44005e7	−1.13487	−0.567435	+	0.823418i	$0.692064\pi$
$858$	0	0
$859$	−1.80489e7	−0.834581	−0.417291	−	0.908773i	$-0.637020\pi$
$859$	−1.80489e7	−0.834581	−0.417291	+	0.908773i	$0.637020\pi$
$860$	0	0
$861$	−196162.	−0.00901792
$862$	0	0
$863$	−2.08506e7	−0.952999	−0.476499	−	0.879175i	$-0.658095\pi$
$863$	−2.08506e7	−0.952999	−0.476499	+	0.879175i	$0.658095\pi$
$864$	0	0
$865$	−2.79152e7	−1.26853
$866$	0	0
$867$	1.00842e6	0.0455612
$868$	0	0
$869$	−2.33047e7	−1.04687
$870$	0	0
$871$	−3.09206e6	−0.138103
$872$	0	0
$873$	2.34224e7	1.04015
$874$	0	0
$875$	−3.90933e6	−0.172616
$876$	0	0
$877$	−1.18628e7	−0.520819	−0.260409	−	0.965498i	$-0.583858\pi$
$877$	−1.18628e7	−0.520819	−0.260409	+	0.965498i	$0.583858\pi$
$878$	0	0
$879$	849628.	0.0370900
$880$	0	0
$881$	−1.67453e7	−0.726866	−0.363433	−	0.931620i	$-0.618395\pi$
$881$	−1.67453e7	−0.726866	−0.363433	+	0.931620i	$0.618395\pi$
$882$	0	0
$883$	3.53639e6	0.152637	0.0763183	−	0.997084i	$-0.475683\pi$
$883$	3.53639e6	0.152637	0.0763183	+	0.997084i	$0.475683\pi$
$884$	0	0
$885$	−2.14866e6	−0.0922169
$886$	0	0
$887$	−3.80164e7	−1.62242	−0.811208	−	0.584758i	$-0.801189\pi$
$887$	−3.80164e7	−1.62242	−0.811208	+	0.584758i	$0.801189\pi$
$888$	0	0
$889$	−5.95787e6	−0.252835
$890$	0	0
$891$	−3.33427e7	−1.40704
$892$	0	0
$893$	−7.05019e7	−2.95850
$894$	0	0
$895$	−5.92950e7	−2.47435
$896$	0	0
$897$	−14173.8	−0.000588174 0
$898$	0	0
$899$	9.23150e6	0.380955
$900$	0	0
$901$	1.56577e7	0.642565
$902$	0	0
$903$	−543094.	−0.0221644
$904$	0	0
$905$	1.79788e7	0.729693
$906$	0	0
$907$	−2.50380e7	−1.01060	−0.505302	−	0.862943i	$-0.668619\pi$
$907$	−2.50380e7	−1.01060	−0.505302	+	0.862943i	$0.668619\pi$
$908$	0	0
$909$	6.45510e6	0.259115
$910$	0	0
$911$	2.36962e7	0.945982	0.472991	−	0.881067i	$-0.343174\pi$
$911$	2.36962e7	0.945982	0.472991	+	0.881067i	$0.343174\pi$
$912$	0	0
$913$	−1.12653e7	−0.447266
$914$	0	0
$915$	220866.	0.00872122
$916$	0	0
$917$	−5.86591e6	−0.230363
$918$	0	0
$919$	−2.73462e7	−1.06809	−0.534044	−	0.845456i	$-0.679329\pi$
$919$	−2.73462e7	−1.06809	−0.534044	+	0.845456i	$0.679329\pi$
$920$	0	0
$921$	1.71437e6	0.0665972
$922$	0	0
$923$	−7.58586e6	−0.293090
$924$	0	0
$925$	7.75231e6	0.297904
$926$	0	0
$927$	1.50816e7	0.576434
$928$	0	0
$929$	3.14288e7	1.19478	0.597390	−	0.801951i	$-0.296204\pi$
$929$	3.14288e7	1.19478	0.597390	+	0.801951i	$0.296204\pi$
$930$	0	0
$931$	−6.12767e6	−0.231697
$932$	0	0
$933$	−1.28771e6	−0.0484300
$934$	0	0
$935$	−1.70815e7	−0.638993
$936$	0	0
$937$	−4.07773e7	−1.51729	−0.758646	−	0.651503i	$-0.774139\pi$
$937$	−4.07773e7	−1.51729	−0.758646	+	0.651503i	$0.774139\pi$
$938$	0	0
$939$	−122213.	−0.00452327
$940$	0	0
$941$	−1.38493e7	−0.509863	−0.254931	−	0.966959i	$-0.582053\pi$
$941$	−1.38493e7	−0.509863	−0.254931	+	0.966959i	$0.582053\pi$
$942$	0	0
$943$	627662.	0.0229851
$944$	0	0
$945$	1.38106e6	0.0503075
$946$	0	0
$947$	1.45927e7	0.528762	0.264381	−	0.964418i	$-0.414832\pi$
$947$	1.45927e7	0.528762	0.264381	+	0.964418i	$0.414832\pi$
$948$	0	0
$949$	9.53679e6	0.343745
$950$	0	0
$951$	−13918.5	−0.000499048 0
$952$	0	0
$953$	2.78156e7	0.992100	0.496050	−	0.868294i	$-0.334783\pi$
$953$	2.78156e7	0.992100	0.496050	+	0.868294i	$0.334783\pi$
$954$	0	0
$955$	−1.19265e7	−0.423161
$956$	0	0
$957$	−1.06660e6	−0.0376462
$958$	0	0
$959$	233622.	0.00820290
$960$	0	0
$961$	−1.26916e7	−0.443312
$962$	0	0
$963$	2.70080e7	0.938484
$964$	0	0
$965$	2.92615e7	1.01153
$966$	0	0
$967$	2.97734e7	1.02391	0.511956	−	0.859012i	$-0.328921\pi$
$967$	2.97734e7	1.02391	0.511956	+	0.859012i	$0.328921\pi$
$968$	0	0
$969$	−865725.	−0.0296190
$970$	0	0
$971$	−4.00707e7	−1.36389	−0.681944	−	0.731404i	$-0.738865\pi$
$971$	−4.00707e7	−1.36389	−0.681944	+	0.731404i	$0.738865\pi$
$972$	0	0
$973$	−8.50055e6	−0.287849
$974$	0	0
$975$	224467.	0.00756209
$976$	0	0
$977$	−3.66964e7	−1.22995	−0.614974	−	0.788547i	$-0.710834\pi$
$977$	−3.66964e7	−1.22995	−0.614974	+	0.788547i	$0.710834\pi$
$978$	0	0
$979$	−5.92888e7	−1.97704
$980$	0	0
$981$	3.95867e7	1.31334
$982$	0	0
$983$	1.14382e7	0.377550	0.188775	−	0.982020i	$-0.439548\pi$
$983$	1.14382e7	0.377550	0.188775	+	0.982020i	$0.439548\pi$
$984$	0	0
$985$	6.41818e7	2.10776
$986$	0	0
$987$	1.09676e6	0.0358360
$988$	0	0
$989$	1.73775e6	0.0564932
$990$	0	0
$991$	−3.22413e7	−1.04286	−0.521432	−	0.853293i	$-0.674602\pi$
$991$	−3.22413e7	−1.04286	−0.521432	+	0.853293i	$0.674602\pi$
$992$	0	0
$993$	2.00483e6	0.0645213
$994$	0	0
$995$	−1.27209e7	−0.407343
$996$	0	0
$997$	−2.17997e7	−0.694566	−0.347283	−	0.937760i	$-0.612896\pi$
$997$	−2.17997e7	−0.694566	−0.347283	+	0.937760i	$0.612896\pi$
$998$	0	0
$999$	1.51533e6	0.0480388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.y.1.1		2
4.3	odd	2		448.6.a.s.1.2		2
8.3	odd	2		224.6.a.d.1.1	yes	2
8.5	even	2		224.6.a.c.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.c.1.2	✓	2	8.5	even	2
224.6.a.d.1.1	yes	2	8.3	odd	2
448.6.a.s.1.2		2	4.3	odd	2
448.6.a.y.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

\(f(q)\)	\(=\)	\(q-0.810250 q^{3} -71.6717 q^{5} -49.0000 q^{7} -242.343 q^{9} -569.271 q^{11} -137.702 q^{13} +58.0720 q^{15} -418.657 q^{17} -2552.13 q^{19} +39.7022 q^{21} -127.036 q^{23} +2011.84 q^{25} +393.249 q^{27} -2312.40 q^{29} -3992.18 q^{31} +461.252 q^{33} +3511.92 q^{35} +3853.34 q^{37} +111.573 q^{39} -4940.82 q^{41} -13679.2 q^{43} +17369.2 q^{45} +27624.7 q^{47} +2401.00 q^{49} +339.216 q^{51} -37400.0 q^{53} +40800.7 q^{55} +2067.86 q^{57} -37000.0 q^{59} +3803.32 q^{61} +11874.8 q^{63} +9869.36 q^{65} +22454.7 q^{67} +102.931 q^{69} +55088.9 q^{71} -69256.6 q^{73} -1630.09 q^{75} +27894.3 q^{77} +40937.8 q^{79} +58570.8 q^{81} +19789.0 q^{83} +30005.8 q^{85} +1873.62 q^{87} +104148. q^{89} +6747.41 q^{91} +3234.66 q^{93} +182916. q^{95} -96649.8 q^{97} +137959. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{3} - 34 q^{5} - 98 q^{7} - 266 q^{9} - 420 q^{11} + 490 q^{13} + 616 q^{15} - 1056 q^{17} - 1246 q^{19} - 686 q^{21} - 504 q^{23} + 306 q^{25} - 3556 q^{27} + 3904 q^{29} + 2044 q^{31} + 2672 q^{33}+ \cdots + 134428 q^{99}+O(q^{100}) \) 2 * q + 14 * q^3 - 34 * q^5 - 98 * q^7 - 266 * q^9 - 420 * q^11 + 490 * q^13 + 616 * q^15 - 1056 * q^17 - 1246 * q^19 - 686 * q^21 - 504 * q^23 + 306 * q^25 - 3556 * q^27 + 3904 * q^29 + 2044 * q^31 + 2672 * q^33 + 1666 * q^35 + 7488 * q^37 + 9408 * q^39 + 7832 * q^41 - 10332 * q^43 + 16478 * q^45 + 41972 * q^47 + 4802 * q^49 - 9100 * q^51 - 32812 * q^53 + 46424 * q^55 + 21412 * q^57 - 48398 * q^59 + 718 * q^61 + 13034 * q^63 + 33516 * q^65 - 12824 * q^67 - 5480 * q^69 + 103992 * q^71 - 54100 * q^73 - 26894 * q^75 + 20580 * q^77 + 64568 * q^79 + 5830 * q^81 + 47810 * q^83 + 5996 * q^85 + 93940 * q^87 - 17388 * q^89 - 24010 * q^91 + 92632 * q^93 + 232120 * q^95 - 97296 * q^97 + 134428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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