LMFDB - Embedded newform 700.6.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,6,Mod(1,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("700.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 700 = 2^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	700.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:	$112.268673869$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 393x^{2} - 1002x + 16596 $ x^4 - 2x^3 - 393x^2 - 1002*x + 16596
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$21.1052$ of defining polynomial
Character	$\chi$	$=$	700.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.1052 q^{3} -49.0000 q^{7} -14.8337 q^{9} +332.519 q^{11} +395.417 q^{13} -15.8479 q^{17} -1508.76 q^{19} -740.153 q^{21} -3178.50 q^{23} -3894.62 q^{27} -4488.23 q^{29} +3120.60 q^{31} +5022.76 q^{33} +15890.5 q^{37} +5972.84 q^{39} +3364.92 q^{41} -102.425 q^{43} -15905.5 q^{47} +2401.00 q^{49} -239.385 q^{51} -40865.2 q^{53} -22790.1 q^{57} -22450.3 q^{59} +24435.9 q^{61} +726.853 q^{63} +2260.27 q^{67} -48011.8 q^{69} +25552.6 q^{71} -25552.9 q^{73} -16293.4 q^{77} -66958.4 q^{79} -55224.4 q^{81} -26284.5 q^{83} -67795.5 q^{87} +40585.1 q^{89} -19375.4 q^{91} +47137.2 q^{93} -57800.0 q^{97} -4932.50 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 22 q^{3} - 196 q^{7} - 62 q^{9} + 48 q^{11} + 430 q^{13} + 1286 q^{17} - 384 q^{19} + 1078 q^{21} - 1256 q^{23} + 1196 q^{27} - 4994 q^{29} + 4618 q^{31} + 15268 q^{33} + 16922 q^{37} + 3200 q^{39}+ \cdots - 170416 q^{99}+O(q^{100}) $ 4 * q - 22 * q^3 - 196 * q^7 - 62 * q^9 + 48 * q^11 + 430 * q^13 + 1286 * q^17 - 384 * q^19 + 1078 * q^21 - 1256 * q^23 + 1196 * q^27 - 4994 * q^29 + 4618 * q^31 + 15268 * q^33 + 16922 * q^37 + 3200 * q^39 - 9192 * q^41 + 12368 * q^43 - 4476 * q^47 + 9604 * q^49 - 30396 * q^51 + 16736 * q^53 - 29946 * q^57 - 14614 * q^59 - 7122 * q^61 + 3038 * q^63 - 21408 * q^67 + 11822 * q^69 - 39748 * q^71 + 89364 * q^73 - 2352 * q^77 - 37640 * q^79 - 125192 * q^81 + 83842 * q^83 - 90982 * q^87 - 12444 * q^89 - 21070 * q^91 + 137840 * q^93 - 57034 * q^97 - 170416 * 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
$3$	15.1052	0.968997	0.484499	−	0.874792i	$-0.339002\pi$
$3$	15.1052	0.968997	0.484499	+	0.874792i	$0.339002\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−14.8337	−0.0610442
$10$	0	0
$11$	332.519	0.828581	0.414290	−	0.910145i	$-0.364030\pi$
$11$	332.519	0.828581	0.414290	+	0.910145i	$0.364030\pi$
$12$	0	0
$13$	395.417	0.648928	0.324464	−	0.945898i	$-0.394816\pi$
$13$	395.417	0.648928	0.324464	+	0.945898i	$0.394816\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−15.8479	−0.0132999	−0.00664996	−	0.999978i	$-0.502117\pi$
$17$	−15.8479	−0.0132999	−0.00664996	+	0.999978i	$0.502117\pi$
$18$	0	0
$19$	−1508.76	−0.958817	−0.479409	−	0.877592i	$-0.659149\pi$
$19$	−1508.76	−0.958817	−0.479409	+	0.877592i	$0.659149\pi$
$20$	0	0
$21$	−740.153	−0.366247
$22$	0	0
$23$	−3178.50	−1.25286	−0.626431	−	0.779477i	$-0.715485\pi$
$23$	−3178.50	−1.25286	−0.626431	+	0.779477i	$0.715485\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3894.62	−1.02815
$28$	0	0
$29$	−4488.23	−0.991015	−0.495508	−	0.868604i	$-0.665018\pi$
$29$	−4488.23	−0.991015	−0.495508	+	0.868604i	$0.665018\pi$
$30$	0	0
$31$	3120.60	0.583222	0.291611	−	0.956537i	$-0.405809\pi$
$31$	3120.60	0.583222	0.291611	+	0.956537i	$0.405809\pi$
$32$	0	0
$33$	5022.76	0.802892
$34$	0	0
$35$	0	0
$36$	0	0
$37$	15890.5	1.90824	0.954118	−	0.299431i	$-0.0967967\pi$
$37$	15890.5	1.90824	0.954118	+	0.299431i	$0.0967967\pi$
$38$	0	0
$39$	5972.84	0.628810
$40$	0	0
$41$	3364.92	0.312619	0.156309	−	0.987708i	$-0.450040\pi$
$41$	3364.92	0.312619	0.156309	+	0.987708i	$0.450040\pi$
$42$	0	0
$43$	−102.425	−0.00844763	−0.00422381	−	0.999991i	$-0.501344\pi$
$43$	−102.425	−0.00844763	−0.00422381	+	0.999991i	$0.501344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−15905.5	−1.05027	−0.525137	−	0.851018i	$-0.675986\pi$
$47$	−15905.5	−1.05027	−0.525137	+	0.851018i	$0.675986\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−239.385	−0.0128876
$52$	0	0
$53$	−40865.2	−1.99832	−0.999158	−	0.0410244i	$-0.986938\pi$
$53$	−40865.2	−1.99832	−0.999158	+	0.0410244i	$0.986938\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−22790.1	−0.929091
$58$	0	0
$59$	−22450.3	−0.839638	−0.419819	−	0.907608i	$-0.637906\pi$
$59$	−22450.3	−0.839638	−0.419819	+	0.907608i	$0.637906\pi$
$60$	0	0
$61$	24435.9	0.840821	0.420411	−	0.907334i	$-0.361886\pi$
$61$	24435.9	0.840821	0.420411	+	0.907334i	$0.361886\pi$
$62$	0	0
$63$	726.853	0.0230725
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2260.27	0.0615139	0.0307569	−	0.999527i	$-0.490208\pi$
$67$	2260.27	0.0615139	0.0307569	+	0.999527i	$0.490208\pi$
$68$	0	0
$69$	−48011.8	−1.21402
$70$	0	0
$71$	25552.6	0.601575	0.300788	−	0.953691i	$-0.402750\pi$
$71$	25552.6	0.601575	0.300788	+	0.953691i	$0.402750\pi$
$72$	0	0
$73$	−25552.9	−0.561219	−0.280610	−	0.959822i	$-0.590537\pi$
$73$	−25552.9	−0.561219	−0.280610	+	0.959822i	$0.590537\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−16293.4	−0.313174
$78$	0	0
$79$	−66958.4	−1.20708	−0.603541	−	0.797332i	$-0.706244\pi$
$79$	−66958.4	−1.20708	−0.603541	+	0.797332i	$0.706244\pi$
$80$	0	0
$81$	−55224.4	−0.935229
$82$	0	0
$83$	−26284.5	−0.418798	−0.209399	−	0.977830i	$-0.567151\pi$
$83$	−26284.5	−0.418798	−0.209399	+	0.977830i	$0.567151\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−67795.5	−0.960291
$88$	0	0
$89$	40585.1	0.543114	0.271557	−	0.962422i	$-0.412461\pi$
$89$	40585.1	0.543114	0.271557	+	0.962422i	$0.412461\pi$
$90$	0	0
$91$	−19375.4	−0.245272
$92$	0	0
$93$	47137.2	0.565141
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−57800.0	−0.623732	−0.311866	−	0.950126i	$-0.600954\pi$
$97$	−57800.0	−0.623732	−0.311866	+	0.950126i	$0.600954\pi$
$98$	0	0
$99$	−4932.50	−0.0505800
$100$	0	0
$101$	−21958.1	−0.214186	−0.107093	−	0.994249i	$-0.534154\pi$
$101$	−21958.1	−0.214186	−0.107093	+	0.994249i	$0.534154\pi$
$102$	0	0
$103$	−39562.2	−0.367441	−0.183720	−	0.982979i	$-0.558814\pi$
$103$	−39562.2	−0.367441	−0.183720	+	0.982979i	$0.558814\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−21102.7	−0.178188	−0.0890941	−	0.996023i	$-0.528397\pi$
$107$	−21102.7	−0.178188	−0.0890941	+	0.996023i	$0.528397\pi$
$108$	0	0
$109$	−221229.	−1.78351	−0.891756	−	0.452517i	$-0.850526\pi$
$109$	−221229.	−1.78351	−0.891756	+	0.452517i	$0.850526\pi$
$110$	0	0
$111$	240028.	1.84908
$112$	0	0
$113$	110973.	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$113$	110973.	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5865.51	−0.0396133
$118$	0	0
$119$	776.547	0.00502690
$120$	0	0
$121$	−50482.1	−0.313454
$122$	0	0
$123$	50827.7	0.302927
$124$	0	0
$125$	0	0
$126$	0	0
$127$	144389.	0.794376	0.397188	−	0.917737i	$-0.369986\pi$
$127$	144389.	0.794376	0.397188	+	0.917737i	$0.369986\pi$
$128$	0	0
$129$	−1547.15	−0.00818573
$130$	0	0
$131$	−275842.	−1.40437	−0.702186	−	0.711994i	$-0.747792\pi$
$131$	−275842.	−1.40437	−0.702186	+	0.711994i	$0.747792\pi$
$132$	0	0
$133$	73929.2	0.362399
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−120194.	−0.547118	−0.273559	−	0.961855i	$-0.588201\pi$
$137$	−120194.	−0.547118	−0.273559	+	0.961855i	$0.588201\pi$
$138$	0	0
$139$	−110020.	−0.482986	−0.241493	−	0.970403i	$-0.577637\pi$
$139$	−110020.	−0.482986	−0.241493	+	0.970403i	$0.577637\pi$
$140$	0	0
$141$	−240255.	−1.01771
$142$	0	0
$143$	131484.	0.537689
$144$	0	0
$145$	0	0
$146$	0	0
$147$	36267.5	0.138428
$148$	0	0
$149$	138049.	0.509412	0.254706	−	0.967019i	$-0.418021\pi$
$149$	138049.	0.509412	0.254706	+	0.967019i	$0.418021\pi$
$150$	0	0
$151$	−73709.8	−0.263077	−0.131538	−	0.991311i	$-0.541992\pi$
$151$	−73709.8	−0.263077	−0.131538	+	0.991311i	$0.541992\pi$
$152$	0	0
$153$	235.084	0.000811883 0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−156298.	−0.506062	−0.253031	−	0.967458i	$-0.581427\pi$
$157$	−156298.	−0.506062	−0.253031	+	0.967458i	$0.581427\pi$
$158$	0	0
$159$	−617276.	−1.93636
$160$	0	0
$161$	155747.	0.473537
$162$	0	0
$163$	437524.	1.28983	0.644916	−	0.764253i	$-0.276892\pi$
$163$	437524.	1.28983	0.644916	+	0.764253i	$0.276892\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−229449.	−0.636642	−0.318321	−	0.947983i	$-0.603119\pi$
$167$	−229449.	−0.636642	−0.318321	+	0.947983i	$0.603119\pi$
$168$	0	0
$169$	−214939.	−0.578892
$170$	0	0
$171$	22380.5	0.0585302
$172$	0	0
$173$	690255.	1.75345	0.876727	−	0.480988i	$-0.159722\pi$
$173$	690255.	1.75345	0.876727	+	0.480988i	$0.159722\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−339116.	−0.813607
$178$	0	0
$179$	−48336.8	−0.112757	−0.0563787	−	0.998409i	$-0.517955\pi$
$179$	−48336.8	−0.112757	−0.0563787	+	0.998409i	$0.517955\pi$
$180$	0	0
$181$	−645264.	−1.46400	−0.732000	−	0.681305i	$-0.761413\pi$
$181$	−645264.	−1.46400	−0.732000	+	0.681305i	$0.761413\pi$
$182$	0	0
$183$	369108.	0.814754
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−5269.72	−0.0110201
$188$	0	0
$189$	190837.	0.388604
$190$	0	0
$191$	−361005.	−0.716028	−0.358014	−	0.933716i	$-0.616546\pi$
$191$	−361005.	−0.716028	−0.358014	+	0.933716i	$0.616546\pi$
$192$	0	0
$193$	187639.	0.362601	0.181301	−	0.983428i	$-0.441969\pi$
$193$	187639.	0.362601	0.181301	+	0.983428i	$0.441969\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−461114.	−0.846531	−0.423265	−	0.906006i	$-0.639116\pi$
$197$	−461114.	−0.846531	−0.423265	+	0.906006i	$0.639116\pi$
$198$	0	0
$199$	−567995.	−1.01675	−0.508373	−	0.861137i	$-0.669753\pi$
$199$	−567995.	−1.01675	−0.508373	+	0.861137i	$0.669753\pi$
$200$	0	0
$201$	34141.7	0.0596068
$202$	0	0
$203$	219923.	0.374569
$204$	0	0
$205$	0	0
$206$	0	0
$207$	47149.1	0.0764799
$208$	0	0
$209$	−501691.	−0.794457
$210$	0	0
$211$	−891786.	−1.37897	−0.689484	−	0.724301i	$-0.742163\pi$
$211$	−891786.	−1.37897	−0.689484	+	0.724301i	$0.742163\pi$
$212$	0	0
$213$	385977.	0.582925
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−152910.	−0.220437
$218$	0	0
$219$	−385981.	−0.543820
$220$	0	0
$221$	−6266.52	−0.00863070
$222$	0	0
$223$	530560.	0.714451	0.357225	−	0.934018i	$-0.383723\pi$
$223$	530560.	0.714451	0.357225	+	0.934018i	$0.383723\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	593649.	0.764655	0.382327	−	0.924027i	$-0.375123\pi$
$227$	593649.	0.764655	0.382327	+	0.924027i	$0.375123\pi$
$228$	0	0
$229$	757698.	0.954789	0.477394	−	0.878689i	$-0.341581\pi$
$229$	757698.	0.954789	0.477394	+	0.878689i	$0.341581\pi$
$230$	0	0
$231$	−246115.	−0.303465
$232$	0	0
$233$	59250.9	0.0714999	0.0357499	−	0.999361i	$-0.488618\pi$
$233$	59250.9	0.0714999	0.0357499	+	0.999361i	$0.488618\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.01142e6	−1.16966
$238$	0	0
$239$	−962045.	−1.08943	−0.544717	−	0.838620i	$-0.683363\pi$
$239$	−962045.	−1.08943	−0.544717	+	0.838620i	$0.683363\pi$
$240$	0	0
$241$	−298605.	−0.331172	−0.165586	−	0.986195i	$-0.552952\pi$
$241$	−298605.	−0.331172	−0.165586	+	0.986195i	$0.552952\pi$
$242$	0	0
$243$	112220.	0.121914
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−596589.	−0.622204
$248$	0	0
$249$	−397032.	−0.405815
$250$	0	0
$251$	943473.	0.945247	0.472624	−	0.881264i	$-0.343307\pi$
$251$	943473.	0.945247	0.472624	+	0.881264i	$0.343307\pi$
$252$	0	0
$253$	−1.05691e6	−1.03810
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−685001.	−0.646931	−0.323466	−	0.946240i	$-0.604848\pi$
$257$	−685001.	−0.646931	−0.323466	+	0.946240i	$0.604848\pi$
$258$	0	0
$259$	−778632.	−0.721245
$260$	0	0
$261$	66577.3	0.0604957
$262$	0	0
$263$	517012.	0.460905	0.230452	−	0.973084i	$-0.425979\pi$
$263$	517012.	0.460905	0.230452	+	0.973084i	$0.425979\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	613044.	0.526276
$268$	0	0
$269$	−2.00093e6	−1.68597	−0.842987	−	0.537934i	$-0.819205\pi$
$269$	−2.00093e6	−1.68597	−0.842987	+	0.537934i	$0.819205\pi$
$270$	0	0
$271$	2.05219e6	1.69744	0.848720	−	0.528842i	$-0.177374\pi$
$271$	2.05219e6	1.69744	0.848720	+	0.528842i	$0.177374\pi$
$272$	0	0
$273$	−292669.	−0.237668
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.03018e6	−0.806700	−0.403350	−	0.915046i	$-0.632154\pi$
$277$	−1.03018e6	−0.806700	−0.403350	+	0.915046i	$0.632154\pi$
$278$	0	0
$279$	−46290.2	−0.0356023
$280$	0	0
$281$	−348383.	−0.263204	−0.131602	−	0.991303i	$-0.542012\pi$
$281$	−348383.	−0.263204	−0.131602	+	0.991303i	$0.542012\pi$
$282$	0	0
$283$	−777820.	−0.577315	−0.288658	−	0.957432i	$-0.593209\pi$
$283$	−777820.	−0.577315	−0.288658	+	0.957432i	$0.593209\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−164881.	−0.118159
$288$	0	0
$289$	−1.41961e6	−0.999823
$290$	0	0
$291$	−873078.	−0.604395
$292$	0	0
$293$	1.57096e6	1.06905	0.534524	−	0.845153i	$-0.320491\pi$
$293$	1.57096e6	1.06905	0.534524	+	0.845153i	$0.320491\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.29504e6	−0.851904
$298$	0	0
$299$	−1.25683e6	−0.813017
$300$	0	0
$301$	5018.82	0.00319290
$302$	0	0
$303$	−331681.	−0.207546
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.47525e6	0.893347	0.446673	−	0.894697i	$-0.352609\pi$
$307$	1.47525e6	0.893347	0.446673	+	0.894697i	$0.352609\pi$
$308$	0	0
$309$	−597594.	−0.356049
$310$	0	0
$311$	−3.31797e6	−1.94523	−0.972617	−	0.232416i	$-0.925337\pi$
$311$	−3.31797e6	−1.94523	−0.972617	+	0.232416i	$0.925337\pi$
$312$	0	0
$313$	94950.8	0.0547820	0.0273910	−	0.999625i	$-0.491280\pi$
$313$	94950.8	0.0547820	0.0273910	+	0.999625i	$0.491280\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−392081.	−0.219143	−0.109572	−	0.993979i	$-0.534948\pi$
$317$	−392081.	−0.219143	−0.109572	+	0.993979i	$0.534948\pi$
$318$	0	0
$319$	−1.49242e6	−0.821136
$320$	0	0
$321$	−318760.	−0.172664
$322$	0	0
$323$	23910.6	0.0127522
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.34170e6	−1.72822
$328$	0	0
$329$	779369.	0.396966
$330$	0	0
$331$	3.72431e6	1.86842	0.934212	−	0.356719i	$-0.116105\pi$
$331$	3.72431e6	1.86842	0.934212	+	0.356719i	$0.116105\pi$
$332$	0	0
$333$	−235715.	−0.116487
$334$	0	0
$335$	0	0
$336$	0	0
$337$	736373.	0.353202	0.176601	−	0.984283i	$-0.443490\pi$
$337$	736373.	0.353202	0.176601	+	0.984283i	$0.443490\pi$
$338$	0	0
$339$	1.67626e6	0.792215
$340$	0	0
$341$	1.03766e6	0.483247
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.84485e6	0.822504	0.411252	−	0.911522i	$-0.365092\pi$
$347$	1.84485e6	0.822504	0.411252	+	0.911522i	$0.365092\pi$
$348$	0	0
$349$	795830.	0.349749	0.174875	−	0.984591i	$-0.444048\pi$
$349$	795830.	0.349749	0.174875	+	0.984591i	$0.444048\pi$
$350$	0	0
$351$	−1.54000e6	−0.667195
$352$	0	0
$353$	−2.87945e6	−1.22991	−0.614954	−	0.788563i	$-0.710826\pi$
$353$	−2.87945e6	−1.22991	−0.614954	+	0.788563i	$0.710826\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11729.9	0.00487105
$358$	0	0
$359$	−137690.	−0.0563854	−0.0281927	−	0.999603i	$-0.508975\pi$
$359$	−137690.	−0.0563854	−0.0281927	+	0.999603i	$0.508975\pi$
$360$	0	0
$361$	−199745.	−0.0806694
$362$	0	0
$363$	−762541.	−0.303736
$364$	0	0
$365$	0	0
$366$	0	0
$367$	621992.	0.241057	0.120528	−	0.992710i	$-0.461541\pi$
$367$	621992.	0.241057	0.120528	+	0.992710i	$0.461541\pi$
$368$	0	0
$369$	−49914.4	−0.0190836
$370$	0	0
$371$	2.00240e6	0.755293
$372$	0	0
$373$	75865.9	0.0282341	0.0141171	−	0.999900i	$-0.495506\pi$
$373$	75865.9	0.0282341	0.0141171	+	0.999900i	$0.495506\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.77472e6	−0.643098
$378$	0	0
$379$	−2.81997e6	−1.00843	−0.504216	−	0.863578i	$-0.668218\pi$
$379$	−2.81997e6	−1.00843	−0.504216	+	0.863578i	$0.668218\pi$
$380$	0	0
$381$	2.18103e6	0.769748
$382$	0	0
$383$	1.06476e6	0.370899	0.185449	−	0.982654i	$-0.440626\pi$
$383$	1.06476e6	0.370899	0.185449	+	0.982654i	$0.440626\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1519.35	0.000515679 0
$388$	0	0
$389$	3.73180e6	1.25039	0.625194	−	0.780469i	$-0.285020\pi$
$389$	3.73180e6	1.25039	0.625194	+	0.780469i	$0.285020\pi$
$390$	0	0
$391$	50372.6	0.0166630
$392$	0	0
$393$	−4.16664e6	−1.36083
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−826618.	−0.263226	−0.131613	−	0.991301i	$-0.542016\pi$
$397$	−826618.	−0.263226	−0.131613	+	0.991301i	$0.542016\pi$
$398$	0	0
$399$	1.11671e6	0.351164
$400$	0	0
$401$	542752.	0.168554	0.0842772	−	0.996442i	$-0.473142\pi$
$401$	542752.	0.168554	0.0842772	+	0.996442i	$0.473142\pi$
$402$	0	0
$403$	1.23394e6	0.378470
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.28388e6	1.58113
$408$	0	0
$409$	3.01368e6	0.890819	0.445409	−	0.895327i	$-0.353058\pi$
$409$	3.01368e6	0.890819	0.445409	+	0.895327i	$0.353058\pi$
$410$	0	0
$411$	−1.81555e6	−0.530156
$412$	0	0
$413$	1.10006e6	0.317353
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.66187e6	−0.468012
$418$	0	0
$419$	3.17741e6	0.884175	0.442088	−	0.896972i	$-0.354238\pi$
$419$	3.17741e6	0.884175	0.442088	+	0.896972i	$0.354238\pi$
$420$	0	0
$421$	−5.99161e6	−1.64755	−0.823774	−	0.566919i	$-0.808135\pi$
$421$	−5.99161e6	−1.64755	−0.823774	+	0.566919i	$0.808135\pi$
$422$	0	0
$423$	235938.	0.0641131
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.19736e6	−0.317801
$428$	0	0
$429$	1.98608e6	0.521020
$430$	0	0
$431$	−934600.	−0.242344	−0.121172	−	0.992632i	$-0.538665\pi$
$431$	−934600.	−0.242344	−0.121172	+	0.992632i	$0.538665\pi$
$432$	0	0
$433$	7.40221e6	1.89733	0.948663	−	0.316289i	$-0.102437\pi$
$433$	7.40221e6	1.89733	0.948663	+	0.316289i	$0.102437\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.79559e6	1.20126
$438$	0	0
$439$	3.34676e6	0.828827	0.414413	−	0.910089i	$-0.363987\pi$
$439$	3.34676e6	0.828827	0.414413	+	0.910089i	$0.363987\pi$
$440$	0	0
$441$	−35615.8	−0.00872060
$442$	0	0
$443$	−1.12131e6	−0.271467	−0.135734	−	0.990745i	$-0.543339\pi$
$443$	−1.12131e6	−0.271467	−0.135734	+	0.990745i	$0.543339\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.08526e6	0.493619
$448$	0	0
$449$	2.30290e6	0.539088	0.269544	−	0.962988i	$-0.413127\pi$
$449$	2.30290e6	0.539088	0.269544	+	0.962988i	$0.413127\pi$
$450$	0	0
$451$	1.11890e6	0.259030
$452$	0	0
$453$	−1.11340e6	−0.254921
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−33146.7	−0.00742420	−0.00371210	−	0.999993i	$-0.501182\pi$
$457$	−33146.7	−0.00742420	−0.00371210	+	0.999993i	$0.501182\pi$
$458$	0	0
$459$	61721.6	0.0136743
$460$	0	0
$461$	6.51273e6	1.42729	0.713643	−	0.700510i	$-0.247044\pi$
$461$	6.51273e6	1.42729	0.713643	+	0.700510i	$0.247044\pi$
$462$	0	0
$463$	−3.05694e6	−0.662726	−0.331363	−	0.943503i	$-0.607508\pi$
$463$	−3.05694e6	−0.662726	−0.331363	+	0.943503i	$0.607508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.90495e6	1.67729	0.838643	−	0.544681i	$-0.183349\pi$
$467$	7.90495e6	1.67729	0.838643	+	0.544681i	$0.183349\pi$
$468$	0	0
$469$	−110753.	−0.0232501
$470$	0	0
$471$	−2.36091e6	−0.490373
$472$	0	0
$473$	−34058.2	−0.00699954
$474$	0	0
$475$	0	0
$476$	0	0
$477$	606184.	0.121986
$478$	0	0
$479$	−8.46668e6	−1.68607	−0.843033	−	0.537862i	$-0.819232\pi$
$479$	−8.46668e6	−1.68607	−0.843033	+	0.537862i	$0.819232\pi$
$480$	0	0
$481$	6.28335e6	1.23831
$482$	0	0
$483$	2.35258e6	0.458856
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.59849e6	1.45179	0.725897	−	0.687803i	$-0.241425\pi$
$487$	7.59849e6	1.45179	0.725897	+	0.687803i	$0.241425\pi$
$488$	0	0
$489$	6.60888e6	1.24984
$490$	0	0
$491$	−7.44368e6	−1.39343	−0.696713	−	0.717350i	$-0.745355\pi$
$491$	−7.44368e6	−1.39343	−0.696713	+	0.717350i	$0.745355\pi$
$492$	0	0
$493$	71129.0	0.0131804
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.25208e6	−0.227374
$498$	0	0
$499$	7.49430e6	1.34735	0.673674	−	0.739029i	$-0.264715\pi$
$499$	7.49430e6	1.34735	0.673674	+	0.739029i	$0.264715\pi$
$500$	0	0
$501$	−3.46587e6	−0.616904
$502$	0	0
$503$	−4.74622e6	−0.836427	−0.418213	−	0.908349i	$-0.637344\pi$
$503$	−4.74622e6	−0.836427	−0.418213	+	0.908349i	$0.637344\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−3.24668e6	−0.560945
$508$	0	0
$509$	7.65723e6	1.31002	0.655009	−	0.755621i	$-0.272665\pi$
$509$	7.65723e6	1.31002	0.655009	+	0.755621i	$0.272665\pi$
$510$	0	0
$511$	1.25209e6	0.212121
$512$	0	0
$513$	5.87605e6	0.985807
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.28888e6	−0.870236
$518$	0	0
$519$	1.04264e7	1.69909
$520$	0	0
$521$	754861.	0.121835	0.0609176	−	0.998143i	$-0.480597\pi$
$521$	754861.	0.121835	0.0609176	+	0.998143i	$0.480597\pi$
$522$	0	0
$523$	182962.	0.0292487	0.0146243	−	0.999893i	$-0.495345\pi$
$523$	182962.	0.0292487	0.0146243	+	0.999893i	$0.495345\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−49455.0	−0.00775682
$528$	0	0
$529$	3.66653e6	0.569661
$530$	0	0
$531$	333022.	0.0512550
$532$	0	0
$533$	1.33055e6	0.202867
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−730135.	−0.109262
$538$	0	0
$539$	798378.	0.118369
$540$	0	0
$541$	1.09713e7	1.61163	0.805813	−	0.592170i	$-0.201729\pi$
$541$	1.09713e7	1.61163	0.805813	+	0.592170i	$0.201729\pi$
$542$	0	0
$543$	−9.74682e6	−1.41861
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.47830e6	1.21155	0.605773	−	0.795637i	$-0.292864\pi$
$547$	8.47830e6	1.21155	0.605773	+	0.795637i	$0.292864\pi$
$548$	0	0
$549$	−362476.	−0.0513273
$550$	0	0
$551$	6.77166e6	0.950202
$552$	0	0
$553$	3.28096e6	0.456234
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.00722e7	1.37558	0.687791	−	0.725909i	$-0.258581\pi$
$557$	1.00722e7	1.37558	0.687791	+	0.725909i	$0.258581\pi$
$558$	0	0
$559$	−40500.6	−0.00548190
$560$	0	0
$561$	−79600.1	−0.0106784
$562$	0	0
$563$	−1.41582e7	−1.88251	−0.941254	−	0.337699i	$-0.890352\pi$
$563$	−1.41582e7	−1.88251	−0.941254	+	0.337699i	$0.890352\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.70599e6	0.353483
$568$	0	0
$569$	2.86677e6	0.371204	0.185602	−	0.982625i	$-0.440577\pi$
$569$	2.86677e6	0.371204	0.185602	+	0.982625i	$0.440577\pi$
$570$	0	0
$571$	−4.30611e6	−0.552707	−0.276354	−	0.961056i	$-0.589126\pi$
$571$	−4.30611e6	−0.552707	−0.276354	+	0.961056i	$0.589126\pi$
$572$	0	0
$573$	−5.45305e6	−0.693829
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.38465e7	1.73141	0.865703	−	0.500558i	$-0.166872\pi$
$577$	1.38465e7	1.73141	0.865703	+	0.500558i	$0.166872\pi$
$578$	0	0
$579$	2.83432e6	0.351360
$580$	0	0
$581$	1.28794e6	0.158291
$582$	0	0
$583$	−1.35885e7	−1.65577
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.82239e6	−0.457867	−0.228934	−	0.973442i	$-0.573524\pi$
$587$	−3.82239e6	−0.457867	−0.228934	+	0.973442i	$0.573524\pi$
$588$	0	0
$589$	−4.70824e6	−0.559204
$590$	0	0
$591$	−6.96521e6	−0.820286
$592$	0	0
$593$	4.69083e6	0.547789	0.273894	−	0.961760i	$-0.411688\pi$
$593$	4.69083e6	0.547789	0.273894	+	0.961760i	$0.411688\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.57967e6	−0.985223
$598$	0	0
$599$	−2.58012e6	−0.293814	−0.146907	−	0.989150i	$-0.546932\pi$
$599$	−2.58012e6	−0.293814	−0.146907	+	0.989150i	$0.546932\pi$
$600$	0	0
$601$	1.23938e7	1.39965	0.699825	−	0.714314i	$-0.253261\pi$
$601$	1.23938e7	1.39965	0.699825	+	0.714314i	$0.253261\pi$
$602$	0	0
$603$	−33528.2	−0.00375506
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.42413e7	−1.56884	−0.784421	−	0.620229i	$-0.787040\pi$
$607$	−1.42413e7	−1.56884	−0.784421	+	0.620229i	$0.787040\pi$
$608$	0	0
$609$	3.32198e6	0.362956
$610$	0	0
$611$	−6.28930e6	−0.681552
$612$	0	0
$613$	−5.67108e6	−0.609557	−0.304779	−	0.952423i	$-0.598583\pi$
$613$	−5.67108e6	−0.609557	−0.304779	+	0.952423i	$0.598583\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.62397e6	0.700495	0.350248	−	0.936657i	$-0.386097\pi$
$617$	6.62397e6	0.700495	0.350248	+	0.936657i	$0.386097\pi$
$618$	0	0
$619$	1.22073e7	1.28054	0.640269	−	0.768151i	$-0.278823\pi$
$619$	1.22073e7	1.28054	0.640269	+	0.768151i	$0.278823\pi$
$620$	0	0
$621$	1.23791e7	1.28813
$622$	0	0
$623$	−1.98867e6	−0.205278
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−7.57813e6	−0.769827
$628$	0	0
$629$	−251830.	−0.0253794
$630$	0	0
$631$	−1.02556e7	−1.02538	−0.512691	−	0.858573i	$-0.671351\pi$
$631$	−1.02556e7	−1.02538	−0.512691	+	0.858573i	$0.671351\pi$
$632$	0	0
$633$	−1.34706e7	−1.33622
$634$	0	0
$635$	0	0
$636$	0	0
$637$	949396.	0.0927041
$638$	0	0
$639$	−379041.	−0.0367227
$640$	0	0
$641$	2.59811e6	0.249754	0.124877	−	0.992172i	$-0.460146\pi$
$641$	2.59811e6	0.249754	0.124877	+	0.992172i	$0.460146\pi$
$642$	0	0
$643$	8.91060e6	0.849923	0.424962	−	0.905211i	$-0.360288\pi$
$643$	8.91060e6	0.849923	0.424962	+	0.905211i	$0.360288\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.22375e7	−1.14929	−0.574647	−	0.818402i	$-0.694860\pi$
$647$	−1.22375e7	−1.14929	−0.574647	+	0.818402i	$0.694860\pi$
$648$	0	0
$649$	−7.46515e6	−0.695708
$650$	0	0
$651$	−2.30972e6	−0.213603
$652$	0	0
$653$	3.70630e6	0.340140	0.170070	−	0.985432i	$-0.445601\pi$
$653$	3.70630e6	0.340140	0.170070	+	0.985432i	$0.445601\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	379045.	0.0342592
$658$	0	0
$659$	−9.92944e6	−0.890658	−0.445329	−	0.895367i	$-0.646913\pi$
$659$	−9.92944e6	−0.890658	−0.445329	+	0.895367i	$0.646913\pi$
$660$	0	0
$661$	1.22697e7	1.09227	0.546137	−	0.837696i	$-0.316098\pi$
$661$	1.22697e7	1.09227	0.546137	+	0.837696i	$0.316098\pi$
$662$	0	0
$663$	−94656.9	−0.00836312
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.42659e7	1.24160
$668$	0	0
$669$	8.01420e6	0.692301
$670$	0	0
$671$	8.12540e6	0.696688
$672$	0	0
$673$	5.84674e6	0.497595	0.248798	−	0.968555i	$-0.419965\pi$
$673$	5.84674e6	0.497595	0.248798	+	0.968555i	$0.419965\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.17692e6	−0.266400	−0.133200	−	0.991089i	$-0.542525\pi$
$677$	−3.17692e6	−0.266400	−0.133200	+	0.991089i	$0.542525\pi$
$678$	0	0
$679$	2.83220e6	0.235749
$680$	0	0
$681$	8.96718e6	0.740948
$682$	0	0
$683$	1.32512e7	1.08693	0.543467	−	0.839431i	$-0.317111\pi$
$683$	1.32512e7	1.08693	0.543467	+	0.839431i	$0.317111\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.14452e7	0.925188
$688$	0	0
$689$	−1.61588e7	−1.29676
$690$	0	0
$691$	−7.46243e6	−0.594546	−0.297273	−	0.954793i	$-0.596077\pi$
$691$	−7.46243e6	−0.594546	−0.297273	+	0.954793i	$0.596077\pi$
$692$	0	0
$693$	241692.	0.0191175
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−53326.9	−0.00415781
$698$	0	0
$699$	894995.	0.0692832
$700$	0	0
$701$	−1.03333e7	−0.794224	−0.397112	−	0.917770i	$-0.629988\pi$
$701$	−1.03333e7	−0.794224	−0.397112	+	0.917770i	$0.629988\pi$
$702$	0	0
$703$	−2.39749e7	−1.82965
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.07595e6	0.0809548
$708$	0	0
$709$	−1.29564e7	−0.967988	−0.483994	−	0.875071i	$-0.660814\pi$
$709$	−1.29564e7	−0.967988	−0.483994	+	0.875071i	$0.660814\pi$
$710$	0	0
$711$	993243.	0.0736854
$712$	0	0
$713$	−9.91884e6	−0.730697
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.45319e7	−1.05566
$718$	0	0
$719$	202068.	0.0145773	0.00728863	−	0.999973i	$-0.497680\pi$
$719$	202068.	0.0145773	0.00728863	+	0.999973i	$0.497680\pi$
$720$	0	0
$721$	1.93855e6	0.138880
$722$	0	0
$723$	−4.51047e6	−0.320905
$724$	0	0
$725$	0	0
$726$	0	0
$727$	8.61730e6	0.604693	0.302347	−	0.953198i	$-0.402230\pi$
$727$	8.61730e6	0.604693	0.302347	+	0.953198i	$0.402230\pi$
$728$	0	0
$729$	1.51146e7	1.05336
$730$	0	0
$731$	1623.22	0.000112353 0
$732$	0	0
$733$	−1.50161e6	−0.103228	−0.0516140	−	0.998667i	$-0.516437\pi$
$733$	−1.50161e6	−0.103228	−0.0516140	+	0.998667i	$0.516437\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	751582.	0.0509692
$738$	0	0
$739$	−1.26282e7	−0.850608	−0.425304	−	0.905051i	$-0.639833\pi$
$739$	−1.26282e7	−0.850608	−0.425304	+	0.905051i	$0.639833\pi$
$740$	0	0
$741$	−9.01158e6	−0.602914
$742$	0	0
$743$	−2.71651e7	−1.80526	−0.902630	−	0.430418i	$-0.858366\pi$
$743$	−2.71651e7	−1.80526	−0.902630	+	0.430418i	$0.858366\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	389898.	0.0255652
$748$	0	0
$749$	1.03403e6	0.0673488
$750$	0	0
$751$	6.85890e6	0.443767	0.221883	−	0.975073i	$-0.428780\pi$
$751$	6.85890e6	0.443767	0.221883	+	0.975073i	$0.428780\pi$
$752$	0	0
$753$	1.42513e7	0.915942
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−5.02373e6	−0.318630	−0.159315	−	0.987228i	$-0.550929\pi$
$757$	−5.02373e6	−0.318630	−0.159315	+	0.987228i	$0.550929\pi$
$758$	0	0
$759$	−1.59648e7	−1.00591
$760$	0	0
$761$	1.11417e7	0.697412	0.348706	−	0.937232i	$-0.386621\pi$
$761$	1.11417e7	0.697412	0.348706	+	0.937232i	$0.386621\pi$
$762$	0	0
$763$	1.08402e7	0.674104
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.87723e6	−0.544865
$768$	0	0
$769$	3.94087e6	0.240312	0.120156	−	0.992755i	$-0.461660\pi$
$769$	3.94087e6	0.240312	0.120156	+	0.992755i	$0.461660\pi$
$770$	0	0
$771$	−1.03471e7	−0.626875
$772$	0	0
$773$	1.57423e7	0.947588	0.473794	−	0.880636i	$-0.342884\pi$
$773$	1.57423e7	0.947588	0.473794	+	0.880636i	$0.342884\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.17614e7	−0.698885
$778$	0	0
$779$	−5.07686e6	−0.299744
$780$	0	0
$781$	8.49674e6	0.498454
$782$	0	0
$783$	1.74800e7	1.01891
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.28875e7	0.741709	0.370854	−	0.928691i	$-0.379065\pi$
$787$	1.28875e7	0.741709	0.370854	+	0.928691i	$0.379065\pi$
$788$	0	0
$789$	7.80955e6	0.446615
$790$	0	0
$791$	−5.43767e6	−0.309009
$792$	0	0
$793$	9.66236e6	0.545633
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.13128e7	1.18849	0.594243	−	0.804286i	$-0.297452\pi$
$797$	2.13128e7	1.18849	0.594243	+	0.804286i	$0.297452\pi$
$798$	0	0
$799$	252069.	0.0139686
$800$	0	0
$801$	−602028.	−0.0331540
$802$	0	0
$803$	−8.49682e6	−0.465015
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.02244e7	−1.63370
$808$	0	0
$809$	1.59293e7	0.855709	0.427854	−	0.903848i	$-0.359270\pi$
$809$	1.59293e7	0.855709	0.427854	+	0.903848i	$0.359270\pi$
$810$	0	0
$811$	−3.28642e7	−1.75457	−0.877286	−	0.479967i	$-0.840648\pi$
$811$	−3.28642e7	−1.75457	−0.877286	+	0.479967i	$0.840648\pi$
$812$	0	0
$813$	3.09987e7	1.64482
$814$	0	0
$815$	0	0
$816$	0	0
$817$	154535.	0.00809973
$818$	0	0
$819$	287410.	0.0149724
$820$	0	0
$821$	2.62592e7	1.35964	0.679820	−	0.733379i	$-0.262058\pi$
$821$	2.62592e7	1.35964	0.679820	+	0.733379i	$0.262058\pi$
$822$	0	0
$823$	2.10397e7	1.08278	0.541388	−	0.840773i	$-0.317899\pi$
$823$	2.10397e7	1.08278	0.541388	+	0.840773i	$0.317899\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.90340e6	0.350994	0.175497	−	0.984480i	$-0.443847\pi$
$827$	6.90340e6	0.350994	0.175497	+	0.984480i	$0.443847\pi$
$828$	0	0
$829$	−2.14263e7	−1.08283	−0.541417	−	0.840754i	$-0.682112\pi$
$829$	−2.14263e7	−1.08283	−0.541417	+	0.840754i	$0.682112\pi$
$830$	0	0
$831$	−1.55610e7	−0.781690
$832$	0	0
$833$	−38050.8	−0.00189999
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.21536e7	−0.599640
$838$	0	0
$839$	−4.08576e6	−0.200386	−0.100193	−	0.994968i	$-0.531946\pi$
$839$	−4.08576e6	−0.200386	−0.100193	+	0.994968i	$0.531946\pi$
$840$	0	0
$841$	−366923.	−0.0178890
$842$	0	0
$843$	−5.26239e6	−0.255044
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.47362e6	0.118475
$848$	0	0
$849$	−1.17491e7	−0.559417
$850$	0	0
$851$	−5.05079e7	−2.39075
$852$	0	0
$853$	3.80900e7	1.79242	0.896208	−	0.443634i	$-0.146311\pi$
$853$	3.80900e7	1.79242	0.896208	+	0.443634i	$0.146311\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.83479e7	−0.853362	−0.426681	−	0.904402i	$-0.640317\pi$
$857$	−1.83479e7	−0.853362	−0.426681	+	0.904402i	$0.640317\pi$
$858$	0	0
$859$	−1.26840e7	−0.586508	−0.293254	−	0.956035i	$-0.594738\pi$
$859$	−1.26840e7	−0.586508	−0.293254	+	0.956035i	$0.594738\pi$
$860$	0	0
$861$	−2.49056e6	−0.114496
$862$	0	0
$863$	4.70445e6	0.215022	0.107511	−	0.994204i	$-0.465712\pi$
$863$	4.70445e6	0.215022	0.107511	+	0.994204i	$0.465712\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.14434e7	−0.968826
$868$	0	0
$869$	−2.22649e7	−1.00017
$870$	0	0
$871$	893748.	0.0399181
$872$	0	0
$873$	857390.	0.0380752
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6.84823e6	−0.300663	−0.150331	−	0.988636i	$-0.548034\pi$
$877$	−6.84823e6	−0.300663	−0.150331	+	0.988636i	$0.548034\pi$
$878$	0	0
$879$	2.37297e7	1.03590
$880$	0	0
$881$	−1.99909e7	−0.867748	−0.433874	−	0.900974i	$-0.642854\pi$
$881$	−1.99909e7	−0.867748	−0.433874	+	0.900974i	$0.642854\pi$
$882$	0	0
$883$	3.08647e7	1.33217	0.666087	−	0.745874i	$-0.267968\pi$
$883$	3.08647e7	1.33217	0.666087	+	0.745874i	$0.267968\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.34136e7	0.572447	0.286224	−	0.958163i	$-0.407600\pi$
$887$	1.34136e7	0.572447	0.286224	+	0.958163i	$0.407600\pi$
$888$	0	0
$889$	−7.07508e6	−0.300246
$890$	0	0
$891$	−1.83631e7	−0.774913
$892$	0	0
$893$	2.39976e7	1.00702
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.89847e7	−0.787811
$898$	0	0
$899$	−1.40060e7	−0.577982
$900$	0	0
$901$	647628.	0.0265775
$902$	0	0
$903$	75810.2	0.00309391
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.38108e7	0.961070	0.480535	−	0.876975i	$-0.340442\pi$
$907$	2.38108e7	0.961070	0.480535	+	0.876975i	$0.340442\pi$
$908$	0	0
$909$	325721.	0.0130748
$910$	0	0
$911$	−1.99857e7	−0.797856	−0.398928	−	0.916982i	$-0.630618\pi$
$911$	−1.99857e7	−0.797856	−0.398928	+	0.916982i	$0.630618\pi$
$912$	0	0
$913$	−8.74011e6	−0.347008
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.35163e7	0.530802
$918$	0	0
$919$	−1.92392e7	−0.751446	−0.375723	−	0.926732i	$-0.622606\pi$
$919$	−1.92392e7	−0.751446	−0.375723	+	0.926732i	$0.622606\pi$
$920$	0	0
$921$	2.22839e7	0.865651
$922$	0	0
$923$	1.01039e7	0.390379
$924$	0	0
$925$	0	0
$926$	0	0
$927$	586855.	0.0224301
$928$	0	0
$929$	−3.35236e7	−1.27442	−0.637208	−	0.770692i	$-0.719911\pi$
$929$	−3.35236e7	−1.27442	−0.637208	+	0.770692i	$0.719911\pi$
$930$	0	0
$931$	−3.62253e6	−0.136974
$932$	0	0
$933$	−5.01185e7	−1.88493
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.49418e7	−1.67225	−0.836125	−	0.548539i	$-0.815184\pi$
$937$	−4.49418e7	−1.67225	−0.836125	+	0.548539i	$0.815184\pi$
$938$	0	0
$939$	1.43425e6	0.0530836
$940$	0	0
$941$	3.75307e7	1.38170	0.690848	−	0.723000i	$-0.257238\pi$
$941$	3.75307e7	1.38170	0.690848	+	0.723000i	$0.257238\pi$
$942$	0	0
$943$	−1.06954e7	−0.391668
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.77961e6	−0.209423	−0.104711	−	0.994503i	$-0.533392\pi$
$947$	−5.77961e6	−0.209423	−0.104711	+	0.994503i	$0.533392\pi$
$948$	0	0
$949$	−1.01040e7	−0.364191
$950$	0	0
$951$	−5.92245e6	−0.212349
$952$	0	0
$953$	4.60167e7	1.64128	0.820642	−	0.571443i	$-0.193616\pi$
$953$	4.60167e7	1.64128	0.820642	+	0.571443i	$0.193616\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.25433e7	−0.795678
$958$	0	0
$959$	5.88950e6	0.206791
$960$	0	0
$961$	−1.88910e7	−0.659852
$962$	0	0
$963$	313032.	0.0108774
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.78796e7	−0.614883	−0.307441	−	0.951567i	$-0.599473\pi$
$967$	−1.78796e7	−0.614883	−0.307441	+	0.951567i	$0.599473\pi$
$968$	0	0
$969$	361175.	0.0123568
$970$	0	0
$971$	−1.22974e7	−0.418568	−0.209284	−	0.977855i	$-0.567113\pi$
$971$	−1.22974e7	−0.418568	−0.209284	+	0.977855i	$0.567113\pi$
$972$	0	0
$973$	5.39097e6	0.182551
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.07260e7	1.02984	0.514920	−	0.857238i	$-0.327822\pi$
$977$	3.07260e7	1.02984	0.514920	+	0.857238i	$0.327822\pi$
$978$	0	0
$979$	1.34953e7	0.450014
$980$	0	0
$981$	3.28165e6	0.108873
$982$	0	0
$983$	−3.81863e7	−1.26044	−0.630222	−	0.776415i	$-0.717036\pi$
$983$	−3.81863e7	−1.26044	−0.630222	+	0.776415i	$0.717036\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.17725e7	0.384659
$988$	0	0
$989$	325558.	0.0105837
$990$	0	0
$991$	−5.23079e6	−0.169193	−0.0845966	−	0.996415i	$-0.526960\pi$
$991$	−5.23079e6	−0.169193	−0.0845966	+	0.996415i	$0.526960\pi$
$992$	0	0
$993$	5.62563e7	1.81050
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.72487e7	−1.50540	−0.752700	−	0.658363i	$-0.771249\pi$
$997$	−4.72487e7	−1.50540	−0.752700	+	0.658363i	$0.771249\pi$
$998$	0	0
$999$	−6.18874e7	−1.96195

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	700.6.a.k.1.4	✓	4
5.2	odd	4		700.6.e.j.449.3		8
5.3	odd	4		700.6.e.j.449.6		8
5.4	even	2		700.6.a.n.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.6.a.k.1.4	✓	4	1.1	even	1	trivial
700.6.a.n.1.1	yes	4	5.4	even	2
700.6.e.j.449.3		8	5.2	odd	4
700.6.e.j.449.6		8	5.3	odd	4

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:	\( N \)	\(=\)	\( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	700.a (trivial)

\(f(q)\)	\(=\)	\(q+15.1052 q^{3} -49.0000 q^{7} -14.8337 q^{9} +332.519 q^{11} +395.417 q^{13} -15.8479 q^{17} -1508.76 q^{19} -740.153 q^{21} -3178.50 q^{23} -3894.62 q^{27} -4488.23 q^{29} +3120.60 q^{31} +5022.76 q^{33} +15890.5 q^{37} +5972.84 q^{39} +3364.92 q^{41} -102.425 q^{43} -15905.5 q^{47} +2401.00 q^{49} -239.385 q^{51} -40865.2 q^{53} -22790.1 q^{57} -22450.3 q^{59} +24435.9 q^{61} +726.853 q^{63} +2260.27 q^{67} -48011.8 q^{69} +25552.6 q^{71} -25552.9 q^{73} -16293.4 q^{77} -66958.4 q^{79} -55224.4 q^{81} -26284.5 q^{83} -67795.5 q^{87} +40585.1 q^{89} -19375.4 q^{91} +47137.2 q^{93} -57800.0 q^{97} -4932.50 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 22 q^{3} - 196 q^{7} - 62 q^{9} + 48 q^{11} + 430 q^{13} + 1286 q^{17} - 384 q^{19} + 1078 q^{21} - 1256 q^{23} + 1196 q^{27} - 4994 q^{29} + 4618 q^{31} + 15268 q^{33} + 16922 q^{37} + 3200 q^{39}+ \cdots - 170416 q^{99}+O(q^{100}) \) 4 * q - 22 * q^3 - 196 * q^7 - 62 * q^9 + 48 * q^11 + 430 * q^13 + 1286 * q^17 - 384 * q^19 + 1078 * q^21 - 1256 * q^23 + 1196 * q^27 - 4994 * q^29 + 4618 * q^31 + 15268 * q^33 + 16922 * q^37 + 3200 * q^39 - 9192 * q^41 + 12368 * q^43 - 4476 * q^47 + 9604 * q^49 - 30396 * q^51 + 16736 * q^53 - 29946 * q^57 - 14614 * q^59 - 7122 * q^61 + 3038 * q^63 - 21408 * q^67 + 11822 * q^69 - 39748 * q^71 + 89364 * q^73 - 2352 * q^77 - 37640 * q^79 - 125192 * q^81 + 83842 * q^83 - 90982 * q^87 - 12444 * q^89 - 21070 * q^91 + 137840 * q^93 - 57034 * q^97 - 170416 * q^99

Introduction

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