LMFDB - Embedded newform 448.6.a.s.1.2

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:	$1$
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			1.2
Root			$-3.40512$ 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	$-0.491727\pi$
$3$	0.810250	0.0519775	0.0259888	+	0.999662i	$0.491727\pi$
$4$	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	$-0.249028\pi$
$11$	569.271	1.41853	0.709264	+	0.704943i	$0.249028\pi$
$12$	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	$-0.198956\pi$
$19$	2552.13	1.62188	0.810941	+	0.585128i	$0.198956\pi$
$20$	0	0
$21$	39.7022	0.0196457
$22$	0	0
$23$	127.036	0.0500734	0.0250367	−	0.999687i	$-0.492030\pi$
$23$	127.036	0.0500734	0.0250367	+	0.999687i	$0.492030\pi$
$24$	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	$-0.378309\pi$
$31$	3992.18	0.746115	0.373058	+	0.927808i	$0.378309\pi$
$32$	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	$-0.309222\pi$
$43$	13679.2	1.12821	0.564103	+	0.825704i	$0.309222\pi$
$44$	0	0
$45$	17369.2	1.27864
$46$	0	0
$47$	−27624.7	−1.82412	−0.912059	−	0.410058i	$-0.865508\pi$
$47$	−27624.7	−1.82412	−0.912059	+	0.410058i	$0.865508\pi$
$48$	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	$-0.256775\pi$
$59$	37000.0	1.38379	0.691897	+	0.721996i	$0.256775\pi$
$60$	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	$-0.598842\pi$
$67$	−22454.7	−0.611111	−0.305555	+	0.952174i	$0.598842\pi$
$68$	0	0
$69$	102.931	0.00260269
$70$	0	0
$71$	−55088.9	−1.29693	−0.648467	−	0.761243i	$-0.724590\pi$
$71$	−55088.9	−1.29693	−0.648467	+	0.761243i	$0.724590\pi$
$72$	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	$-0.620300\pi$
$79$	−40937.8	−0.738000	−0.369000	+	0.929429i	$0.620300\pi$
$80$	0	0
$81$	58570.8	0.991902
$82$	0	0
$83$	−19789.0	−0.315303	−0.157652	−	0.987495i	$-0.550392\pi$
$83$	−19789.0	−0.315303	−0.157652	+	0.987495i	$0.550392\pi$
$84$	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	$-0.406678\pi$
$103$	62232.5	0.577995	0.288998	+	0.957330i	$0.406678\pi$
$104$	0	0
$105$	−2845.53	−0.0251878
$106$	0	0
$107$	111445.	0.941026	0.470513	−	0.882393i	$-0.344069\pi$
$107$	111445.	0.941026	0.470513	+	0.882393i	$0.344069\pi$
$108$	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	$-0.608557\pi$
$127$	−121589.	−0.668938	−0.334469	+	0.942407i	$0.608557\pi$
$128$	0	0
$129$	11083.5	0.0586414
$130$	0	0
$131$	−119712.	−0.609482	−0.304741	−	0.952435i	$-0.598570\pi$
$131$	−119712.	−0.609482	−0.304741	+	0.952435i	$0.598570\pi$
$132$	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	$-0.624348\pi$
$139$	−173481.	−0.761578	−0.380789	+	0.924662i	$0.624348\pi$
$140$	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	$-0.332815\pi$
$151$	280973.	1.00282	0.501410	+	0.865210i	$0.332815\pi$
$152$	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	$-0.880672\pi$
$163$	−631306.	−1.86110	−0.930552	+	0.366160i	$0.880672\pi$
$164$	0	0
$165$	−33058.7	−0.0945315
$166$	0	0
$167$	−174139.	−0.483176	−0.241588	−	0.970379i	$-0.577668\pi$
$167$	−174139.	−0.483176	−0.241588	+	0.970379i	$0.577668\pi$
$168$	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	$-0.915482\pi$
$179$	−827314.	−1.92991	−0.964956	+	0.262411i	$0.915482\pi$
$180$	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	$-0.552771\pi$
$191$	−166405.	−0.330052	−0.165026	+	0.986289i	$0.552771\pi$
$192$	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	$-0.550781\pi$
$199$	−177488.	−0.317714	−0.158857	+	0.987302i	$0.550781\pi$
$200$	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	$-0.232496\pi$
$211$	963464.	1.48981	0.744903	+	0.667173i	$0.232496\pi$
$212$	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	$-0.387622\pi$
$223$	513529.	0.691518	0.345759	+	0.938323i	$0.387622\pi$
$224$	0	0
$225$	−487556.	−0.642049
$226$	0	0
$227$	−251707.	−0.324213	−0.162106	−	0.986773i	$-0.551829\pi$
$227$	−251707.	−0.324213	−0.162106	+	0.986773i	$0.551829\pi$
$228$	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	$-0.131440\pi$
$239$	1.61769e6	1.83190	0.915949	+	0.401295i	$0.131440\pi$
$240$	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	$-0.521166\pi$
$251$	−132644.	−0.132893	−0.0664466	+	0.997790i	$0.521166\pi$
$252$	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	$-0.419764\pi$
$263$	559540.	0.498818	0.249409	+	0.968398i	$0.419764\pi$
$264$	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	$-0.222208\pi$
$271$	1.85235e6	1.53214	0.766072	+	0.642754i	$0.222208\pi$
$272$	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	$-0.253906\pi$
$283$	1.88185e6	1.39675	0.698376	+	0.715731i	$0.253906\pi$
$284$	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	$-0.278671\pi$
$307$	2.11586e6	1.28127	0.640634	+	0.767846i	$0.278671\pi$
$308$	0	0
$309$	50423.9	0.0300428
$310$	0	0
$311$	−1.58928e6	−0.931749	−0.465875	−	0.884851i	$-0.654260\pi$
$311$	−1.58928e6	−0.931749	−0.465875	+	0.884851i	$0.654260\pi$
$312$	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	$-0.286862\pi$
$331$	2.47433e6	1.24133	0.620666	+	0.784075i	$0.286862\pi$
$332$	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	$-0.540310\pi$
$347$	−566575.	−0.252600	−0.126300	+	0.991992i	$0.540310\pi$
$348$	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	$-0.299111\pi$
$359$	2.88170e6	1.18008	0.590042	+	0.807373i	$0.299111\pi$
$360$	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	$-0.217211\pi$
$367$	4.00494e6	1.55214	0.776070	+	0.630647i	$0.217211\pi$
$368$	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	$-0.455978\pi$
$379$	771013.	0.275717	0.137858	+	0.990452i	$0.455978\pi$
$380$	0	0
$381$	−98517.6	−0.0347697
$382$	0	0
$383$	1.49145e6	0.519530	0.259765	−	0.965672i	$-0.416355\pi$
$383$	1.49145e6	0.519530	0.259765	+	0.965672i	$0.416355\pi$
$384$	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	$-0.748447\pi$
$419$	−5.05733e6	−1.40730	−0.703649	+	0.710548i	$0.748447\pi$
$420$	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	$-0.279555\pi$
$431$	4.92475e6	1.27700	0.638500	+	0.769622i	$0.279555\pi$
$432$	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	$-0.717312\pi$
$439$	−5.09504e6	−1.26179	−0.630894	+	0.775869i	$0.717312\pi$
$440$	0	0
$441$	−581867.	−0.142471
$442$	0	0
$443$	2.63893e6	0.638878	0.319439	−	0.947607i	$-0.396506\pi$
$443$	2.63893e6	0.638878	0.319439	+	0.947607i	$0.396506\pi$
$444$	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	$-0.437756\pi$
$463$	1.79250e6	0.388603	0.194301	+	0.980942i	$0.437756\pi$
$464$	0	0
$465$	−231834.	−0.0497215
$466$	0	0
$467$	−5.56530e6	−1.18085	−0.590427	−	0.807091i	$-0.701041\pi$
$467$	−5.56530e6	−1.18085	−0.590427	+	0.807091i	$0.701041\pi$
$468$	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	$-0.792472\pi$
$479$	−7.98318e6	−1.58978	−0.794890	+	0.606753i	$0.792472\pi$
$480$	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	$-0.557874\pi$
$487$	−1.89273e6	−0.361631	−0.180815	+	0.983517i	$0.557874\pi$
$488$	0	0
$489$	−511515.	−0.0967356
$490$	0	0
$491$	714559.	0.133762	0.0668812	−	0.997761i	$-0.478695\pi$
$491$	714559.	0.133762	0.0668812	+	0.997761i	$0.478695\pi$
$492$	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	$-0.700481\pi$
$499$	−6.55243e6	−1.17802	−0.589008	+	0.808127i	$0.700481\pi$
$500$	0	0
$501$	−141096.	−0.0251143
$502$	0	0
$503$	−990048.	−0.174476	−0.0872381	−	0.996187i	$-0.527804\pi$
$503$	−990048.	−0.174476	−0.0872381	+	0.996187i	$0.527804\pi$
$504$	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	$-0.625755\pi$
$523$	−4.81508e6	−0.769749	−0.384875	+	0.922969i	$0.625755\pi$
$524$	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	$-0.640281\pi$
$547$	−5.97032e6	−0.853158	−0.426579	+	0.904450i	$0.640281\pi$
$548$	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	$-0.766544\pi$
$563$	−1.11744e7	−1.48578	−0.742888	+	0.669416i	$0.766544\pi$
$564$	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	$-0.277250\pi$
$571$	1.00356e7	1.28811	0.644057	+	0.764977i	$0.277250\pi$
$572$	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	$-0.439356\pi$
$587$	3.16178e6	0.378736	0.189368	+	0.981906i	$0.439356\pi$
$588$	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	$-0.850570\pi$
$599$	−1.56629e7	−1.78364	−0.891818	+	0.452394i	$0.850570\pi$
$600$	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	$-0.739568\pi$
$607$	−1.24101e7	−1.36711	−0.683557	+	0.729897i	$0.739568\pi$
$608$	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	$-0.430180\pi$
$619$	4.14856e6	0.435181	0.217591	+	0.976040i	$0.430180\pi$
$620$	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	$-0.812553\pi$
$631$	−1.66340e7	−1.66312	−0.831561	+	0.555433i	$0.812553\pi$
$632$	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	$-0.262257\pi$
$643$	1.42448e7	1.35872	0.679360	+	0.733805i	$0.262257\pi$
$644$	0	0
$645$	−794377.	−0.0751843
$646$	0	0
$647$	−9.78031e6	−0.918527	−0.459264	−	0.888300i	$-0.651887\pi$
$647$	−9.78031e6	−0.918527	−0.459264	+	0.888300i	$0.651887\pi$
$648$	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	$-0.550032\pi$
$659$	−3.49022e6	−0.313069	−0.156534	+	0.987673i	$0.550032\pi$
$660$	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	$-0.773090\pi$
$683$	−1.84454e7	−1.51299	−0.756495	+	0.654000i	$0.773090\pi$
$684$	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	$-0.429386\pi$
$691$	5.52332e6	0.440053	0.220027	+	0.975494i	$0.429386\pi$
$692$	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	$-0.471078\pi$
$719$	2.51554e6	0.181472	0.0907360	+	0.995875i	$0.471078\pi$
$720$	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	$-0.683038\pi$
$727$	−1.55008e7	−1.08772	−0.543862	+	0.839175i	$0.683038\pi$
$728$	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	$-0.470663\pi$
$739$	2.73270e6	0.184069	0.0920346	+	0.995756i	$0.470663\pi$
$740$	0	0
$741$	−284749.	−0.0190510
$742$	0	0
$743$	−108079.	−0.00718239	−0.00359119	−	0.999994i	$-0.501143\pi$
$743$	−108079.	−0.00718239	−0.00359119	+	0.999994i	$0.501143\pi$
$744$	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	$-0.651531\pi$
$751$	−1.41662e7	−0.916543	−0.458272	+	0.888812i	$0.651531\pi$
$752$	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	$-0.0887360\pi$
$787$	3.34094e7	1.92279	0.961394	+	0.275176i	$0.0887360\pi$
$788$	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	$-0.542109\pi$
$811$	−4.94124e6	−0.263805	−0.131903	+	0.991263i	$0.542109\pi$
$812$	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	$-0.575386\pi$
$823$	−9.11809e6	−0.469250	−0.234625	+	0.972086i	$0.575386\pi$
$824$	0	0
$825$	927965.	0.0474675
$826$	0	0
$827$	−8.21913e6	−0.417890	−0.208945	−	0.977927i	$-0.567003\pi$
$827$	−8.21913e6	−0.417890	−0.208945	+	0.977927i	$0.567003\pi$
$828$	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	$-0.464402\pi$
$839$	4.55100e6	0.223204	0.111602	+	0.993753i	$0.464402\pi$
$840$	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	$-0.362980\pi$
$859$	1.80489e7	0.834581	0.417291	+	0.908773i	$0.362980\pi$
$860$	0	0
$861$	−196162.	−0.00901792
$862$	0	0
$863$	2.08506e7	0.952999	0.476499	−	0.879175i	$-0.341905\pi$
$863$	2.08506e7	0.952999	0.476499	+	0.879175i	$0.341905\pi$
$864$	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	$-0.524317\pi$
$883$	−3.53639e6	−0.152637	−0.0763183	+	0.997084i	$0.524317\pi$
$884$	0	0
$885$	−2.14866e6	−0.0922169
$886$	0	0
$887$	3.80164e7	1.62242	0.811208	−	0.584758i	$-0.198811\pi$
$887$	3.80164e7	1.62242	0.811208	+	0.584758i	$0.198811\pi$
$888$	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	$-0.331381\pi$
$907$	2.50380e7	1.01060	0.505302	+	0.862943i	$0.331381\pi$
$908$	0	0
$909$	6.45510e6	0.259115
$910$	0	0
$911$	−2.36962e7	−0.945982	−0.472991	−	0.881067i	$-0.656826\pi$
$911$	−2.36962e7	−0.945982	−0.472991	+	0.881067i	$0.656826\pi$
$912$	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	$-0.320671\pi$
$919$	2.73462e7	1.06809	0.534044	+	0.845456i	$0.320671\pi$
$920$	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	$-0.585168\pi$
$947$	−1.45927e7	−0.528762	−0.264381	+	0.964418i	$0.585168\pi$
$948$	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	$-0.671079\pi$
$967$	−2.97734e7	−1.02391	−0.511956	+	0.859012i	$0.671079\pi$
$968$	0	0
$969$	−865725.	−0.0296190
$970$	0	0
$971$	4.00707e7	1.36389	0.681944	−	0.731404i	$-0.261135\pi$
$971$	4.00707e7	1.36389	0.681944	+	0.731404i	$0.261135\pi$
$972$	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	$-0.560452\pi$
$983$	−1.14382e7	−0.377550	−0.188775	+	0.982020i	$0.560452\pi$
$984$	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	$-0.325398\pi$
$991$	3.22413e7	1.04286	0.521432	+	0.853293i	$0.325398\pi$
$992$	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.s.1.2		2
4.3	odd	2		448.6.a.y.1.1		2
8.3	odd	2		224.6.a.c.1.2	✓	2
8.5	even	2		224.6.a.d.1.1	yes	2

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
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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