LMFDB - Embedded newform 704.6.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,6,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	704.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:	$112.910209148$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{31}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 31 $ x^2 - 31
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 44)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.56776$ of defining polynomial
Character	$\chi$	$=$	704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+25.2711 q^{3} +55.5421 q^{5} +200.813 q^{7} +395.626 q^{9} +121.000 q^{11} -727.355 q^{13} +1403.61 q^{15} -1108.74 q^{17} -1662.40 q^{19} +5074.76 q^{21} +2986.99 q^{23} -40.0735 q^{25} +3857.03 q^{27} +1550.86 q^{29} +9511.56 q^{31} +3057.80 q^{33} +11153.6 q^{35} +9430.48 q^{37} -18381.0 q^{39} +7371.29 q^{41} +8528.56 q^{43} +21973.9 q^{45} +30033.5 q^{47} +23518.9 q^{49} -28019.0 q^{51} -23965.7 q^{53} +6720.60 q^{55} -42010.7 q^{57} +6965.57 q^{59} -49080.3 q^{61} +79447.0 q^{63} -40398.9 q^{65} +23928.7 q^{67} +75484.5 q^{69} -1881.22 q^{71} -13674.6 q^{73} -1012.70 q^{75} +24298.4 q^{77} +11584.9 q^{79} +1334.00 q^{81} +55143.4 q^{83} -61581.7 q^{85} +39191.8 q^{87} -2372.81 q^{89} -146063. q^{91} +240367. q^{93} -92333.4 q^{95} -7919.13 q^{97} +47870.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 22 q^{5} + 268 q^{7} + 524 q^{9} + 242 q^{11} - 1232 q^{13} + 2050 q^{15} - 124 q^{17} - 1944 q^{19} + 3780 q^{21} + 3346 q^{23} - 2040 q^{25} + 6066 q^{27} + 6576 q^{29} + 2498 q^{31} + 726 q^{33}+ \cdots + 63404 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 22 * q^5 + 268 * q^7 + 524 * q^9 + 242 * q^11 - 1232 * q^13 + 2050 * q^15 - 124 * q^17 - 1944 * q^19 + 3780 * q^21 + 3346 * q^23 - 2040 * q^25 + 6066 * q^27 + 6576 * q^29 + 2498 * q^31 + 726 * q^33 + 8900 * q^35 + 14674 * q^37 - 8656 * q^39 - 6504 * q^41 - 11628 * q^43 + 17668 * q^45 + 36816 * q^47 + 11226 * q^49 - 46996 * q^51 + 3292 * q^53 + 2662 * q^55 - 36584 * q^57 - 12126 * q^59 - 73128 * q^61 + 88072 * q^63 - 23472 * q^65 - 29334 * q^67 + 68566 * q^69 - 46122 * q^71 + 8240 * q^73 + 37528 * q^75 + 32428 * q^77 - 14780 * q^79 - 72430 * q^81 + 74564 * q^83 - 94612 * q^85 - 57648 * q^87 - 33698 * q^89 - 179968 * q^91 + 375526 * q^93 - 82888 * q^95 + 126162 * q^97 + 63404 * 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
$3$	25.2711	1.62114	0.810570	−	0.585642i	$-0.199158\pi$
$3$	25.2711	1.62114	0.810570	+	0.585642i	$0.199158\pi$
$4$	0	0
$5$	55.5421	0.993568	0.496784	−	0.867874i	$-0.334514\pi$
$5$	55.5421	0.993568	0.496784	+	0.867874i	$0.334514\pi$
$6$	0	0
$7$	200.813	1.54898	0.774492	−	0.632583i	$-0.218005\pi$
$7$	200.813	1.54898	0.774492	+	0.632583i	$0.218005\pi$
$8$	0	0
$9$	395.626	1.62809
$10$	0	0
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	−727.355	−1.19368	−0.596840	−	0.802360i	$-0.703577\pi$
$13$	−727.355	−1.19368	−0.596840	+	0.802360i	$0.703577\pi$
$14$	0	0
$15$	1403.61	1.61071
$16$	0	0
$17$	−1108.74	−0.930481	−0.465240	−	0.885184i	$-0.654032\pi$
$17$	−1108.74	−0.930481	−0.465240	+	0.885184i	$0.654032\pi$
$18$	0	0
$19$	−1662.40	−1.05646	−0.528229	−	0.849102i	$-0.677144\pi$
$19$	−1662.40	−1.05646	−0.528229	+	0.849102i	$0.677144\pi$
$20$	0	0
$21$	5074.76	2.51112
$22$	0	0
$23$	2986.99	1.17737	0.588687	−	0.808361i	$-0.299645\pi$
$23$	2986.99	1.17737	0.588687	+	0.808361i	$0.299645\pi$
$24$	0	0
$25$	−40.0735	−0.0128235
$26$	0	0
$27$	3857.03	1.01822
$28$	0	0
$29$	1550.86	0.342434	0.171217	−	0.985233i	$-0.445230\pi$
$29$	1550.86	0.342434	0.171217	+	0.985233i	$0.445230\pi$
$30$	0	0
$31$	9511.56	1.77766	0.888828	−	0.458241i	$-0.151520\pi$
$31$	9511.56	1.77766	0.888828	+	0.458241i	$0.151520\pi$
$32$	0	0
$33$	3057.80	0.488792
$34$	0	0
$35$	11153.6	1.53902
$36$	0	0
$37$	9430.48	1.13248	0.566239	−	0.824241i	$-0.308398\pi$
$37$	9430.48	1.13248	0.566239	+	0.824241i	$0.308398\pi$
$38$	0	0
$39$	−18381.0	−1.93512
$40$	0	0
$41$	7371.29	0.684832	0.342416	−	0.939548i	$-0.388755\pi$
$41$	7371.29	0.684832	0.342416	+	0.939548i	$0.388755\pi$
$42$	0	0
$43$	8528.56	0.703404	0.351702	−	0.936112i	$-0.385603\pi$
$43$	8528.56	0.703404	0.351702	+	0.936112i	$0.385603\pi$
$44$	0	0
$45$	21973.9	1.61762
$46$	0	0
$47$	30033.5	1.98318	0.991588	−	0.129436i	$-0.0413165\pi$
$47$	30033.5	1.98318	0.991588	+	0.129436i	$0.0413165\pi$
$48$	0	0
$49$	23518.9	1.39935
$50$	0	0
$51$	−28019.0	−1.50844
$52$	0	0
$53$	−23965.7	−1.17193	−0.585964	−	0.810337i	$-0.699284\pi$
$53$	−23965.7	−1.17193	−0.585964	+	0.810337i	$0.699284\pi$
$54$	0	0
$55$	6720.60	0.299572
$56$	0	0
$57$	−42010.7	−1.71267
$58$	0	0
$59$	6965.57	0.260511	0.130256	−	0.991480i	$-0.458420\pi$
$59$	6965.57	0.260511	0.130256	+	0.991480i	$0.458420\pi$
$60$	0	0
$61$	−49080.3	−1.68882	−0.844409	−	0.535699i	$-0.820048\pi$
$61$	−49080.3	−1.68882	−0.844409	+	0.535699i	$0.820048\pi$
$62$	0	0
$63$	79447.0	2.52189
$64$	0	0
$65$	−40398.9	−1.18600
$66$	0	0
$67$	23928.7	0.651228	0.325614	−	0.945503i	$-0.394429\pi$
$67$	23928.7	0.651228	0.325614	+	0.945503i	$0.394429\pi$
$68$	0	0
$69$	75484.5	1.90869
$70$	0	0
$71$	−1881.22	−0.0442889	−0.0221444	−	0.999755i	$-0.507049\pi$
$71$	−1881.22	−0.0442889	−0.0221444	+	0.999755i	$0.507049\pi$
$72$	0	0
$73$	−13674.6	−0.300336	−0.150168	−	0.988661i	$-0.547981\pi$
$73$	−13674.6	−0.300336	−0.150168	+	0.988661i	$0.547981\pi$
$74$	0	0
$75$	−1012.70	−0.0207887
$76$	0	0
$77$	24298.4	0.467036
$78$	0	0
$79$	11584.9	0.208846	0.104423	−	0.994533i	$-0.466700\pi$
$79$	11584.9	0.208846	0.104423	+	0.994533i	$0.466700\pi$
$80$	0	0
$81$	1334.00	0.0225915
$82$	0	0
$83$	55143.4	0.878614	0.439307	−	0.898337i	$-0.355224\pi$
$83$	55143.4	0.878614	0.439307	+	0.898337i	$0.355224\pi$
$84$	0	0
$85$	−61581.7	−0.924495
$86$	0	0
$87$	39191.8	0.555133
$88$	0	0
$89$	−2372.81	−0.0317533	−0.0158766	−	0.999874i	$-0.505054\pi$
$89$	−2372.81	−0.0317533	−0.0158766	+	0.999874i	$0.505054\pi$
$90$	0	0
$91$	−146063.	−1.84899
$92$	0	0
$93$	240367.	2.88183
$94$	0	0
$95$	−92333.4	−1.04966
$96$	0	0
$97$	−7919.13	−0.0854571	−0.0427286	−	0.999087i	$-0.513605\pi$
$97$	−7919.13	−0.0854571	−0.0427286	+	0.999087i	$0.513605\pi$
$98$	0	0
$99$	47870.8	0.490888
$100$	0	0
$101$	45021.1	0.439150	0.219575	−	0.975596i	$-0.429533\pi$
$101$	45021.1	0.439150	0.219575	+	0.975596i	$0.429533\pi$
$102$	0	0
$103$	−112625.	−1.04603	−0.523013	−	0.852325i	$-0.675192\pi$
$103$	−112625.	−1.04603	−0.523013	+	0.852325i	$0.675192\pi$
$104$	0	0
$105$	281863.	2.49497
$106$	0	0
$107$	207255.	1.75003	0.875015	−	0.484096i	$-0.160851\pi$
$107$	207255.	1.75003	0.875015	+	0.484096i	$0.160851\pi$
$108$	0	0
$109$	−94995.8	−0.765840	−0.382920	−	0.923781i	$-0.625082\pi$
$109$	−94995.8	−0.765840	−0.382920	+	0.923781i	$0.625082\pi$
$110$	0	0
$111$	238318.	1.83590
$112$	0	0
$113$	84333.5	0.621304	0.310652	−	0.950524i	$-0.399453\pi$
$113$	84333.5	0.621304	0.310652	+	0.950524i	$0.399453\pi$
$114$	0	0
$115$	165904.	1.16980
$116$	0	0
$117$	−287761.	−1.94342
$118$	0	0
$119$	−222650.	−1.44130
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	186280.	1.11021
$124$	0	0
$125$	−175795.	−1.00631
$126$	0	0
$127$	33416.9	0.183847	0.0919236	−	0.995766i	$-0.470698\pi$
$127$	33416.9	0.183847	0.0919236	+	0.995766i	$0.470698\pi$
$128$	0	0
$129$	215526.	1.14032
$130$	0	0
$131$	−353269.	−1.79857	−0.899286	−	0.437362i	$-0.855913\pi$
$131$	−353269.	−1.79857	−0.899286	+	0.437362i	$0.855913\pi$
$132$	0	0
$133$	−333832.	−1.63644
$134$	0	0
$135$	214228.	1.01167
$136$	0	0
$137$	−195043.	−0.887829	−0.443915	−	0.896069i	$-0.646411\pi$
$137$	−195043.	−0.887829	−0.443915	+	0.896069i	$0.646411\pi$
$138$	0	0
$139$	−297644.	−1.30665	−0.653326	−	0.757077i	$-0.726627\pi$
$139$	−297644.	−1.30665	−0.653326	+	0.757077i	$0.726627\pi$
$140$	0	0
$141$	758978.	3.21500
$142$	0	0
$143$	−88010.0	−0.359908
$144$	0	0
$145$	86137.9	0.340231
$146$	0	0
$147$	594348.	2.26855
$148$	0	0
$149$	−32976.3	−0.121685	−0.0608423	−	0.998147i	$-0.519379\pi$
$149$	−32976.3	−0.121685	−0.0608423	+	0.998147i	$0.519379\pi$
$150$	0	0
$151$	325852.	1.16299	0.581497	−	0.813548i	$-0.302467\pi$
$151$	325852.	1.16299	0.581497	+	0.813548i	$0.302467\pi$
$152$	0	0
$153$	−438647.	−1.51491
$154$	0	0
$155$	528292.	1.76622
$156$	0	0
$157$	362988.	1.17528	0.587642	−	0.809121i	$-0.300056\pi$
$157$	362988.	1.17528	0.587642	+	0.809121i	$0.300056\pi$
$158$	0	0
$159$	−605639.	−1.89986
$160$	0	0
$161$	599827.	1.82373
$162$	0	0
$163$	−597696.	−1.76202	−0.881011	−	0.473095i	$-0.843137\pi$
$163$	−597696.	−1.76202	−0.881011	+	0.473095i	$0.843137\pi$
$164$	0	0
$165$	169837.	0.485648
$166$	0	0
$167$	−474413.	−1.31633	−0.658166	−	0.752873i	$-0.728667\pi$
$167$	−474413.	−1.31633	−0.658166	+	0.752873i	$0.728667\pi$
$168$	0	0
$169$	157753.	0.424874
$170$	0	0
$171$	−657690.	−1.72001
$172$	0	0
$173$	−48727.9	−0.123783	−0.0618917	−	0.998083i	$-0.519713\pi$
$173$	−48727.9	−0.123783	−0.0618917	+	0.998083i	$0.519713\pi$
$174$	0	0
$175$	−8047.28	−0.0198634
$176$	0	0
$177$	176027.	0.422325
$178$	0	0
$179$	−615070.	−1.43480	−0.717401	−	0.696660i	$-0.754668\pi$
$179$	−615070.	−1.43480	−0.717401	+	0.696660i	$0.754668\pi$
$180$	0	0
$181$	−101539.	−0.230374	−0.115187	−	0.993344i	$-0.536747\pi$
$181$	−101539.	−0.230374	−0.115187	+	0.993344i	$0.536747\pi$
$182$	0	0
$183$	−1.24031e6	−2.73781
$184$	0	0
$185$	523789.	1.12519
$186$	0	0
$187$	−134158.	−0.280550
$188$	0	0
$189$	774542.	1.57721
$190$	0	0
$191$	41311.8	0.0819389	0.0409695	−	0.999160i	$-0.486955\pi$
$191$	41311.8	0.0819389	0.0409695	+	0.999160i	$0.486955\pi$
$192$	0	0
$193$	−487524.	−0.942111	−0.471056	−	0.882103i	$-0.656127\pi$
$193$	−487524.	−0.942111	−0.471056	+	0.882103i	$0.656127\pi$
$194$	0	0
$195$	−1.02092e6	−1.92268
$196$	0	0
$197$	393575.	0.722539	0.361270	−	0.932461i	$-0.382343\pi$
$197$	393575.	0.722539	0.361270	+	0.932461i	$0.382343\pi$
$198$	0	0
$199$	−295187.	−0.528401	−0.264201	−	0.964468i	$-0.585108\pi$
$199$	−295187.	−0.528401	−0.264201	+	0.964468i	$0.585108\pi$
$200$	0	0
$201$	604705.	1.05573
$202$	0	0
$203$	311433.	0.530425
$204$	0	0
$205$	409417.	0.680427
$206$	0	0
$207$	1.18173e6	1.91687
$208$	0	0
$209$	−201151.	−0.318534
$210$	0	0
$211$	481974.	0.745276	0.372638	−	0.927977i	$-0.378453\pi$
$211$	481974.	0.745276	0.372638	+	0.927977i	$0.378453\pi$
$212$	0	0
$213$	−47540.5	−0.0717984
$214$	0	0
$215$	473694.	0.698879
$216$	0	0
$217$	1.91005e6	2.75356
$218$	0	0
$219$	−345571.	−0.486886
$220$	0	0
$221$	806448.	1.11070
$222$	0	0
$223$	233896.	0.314964	0.157482	−	0.987522i	$-0.449662\pi$
$223$	233896.	0.314964	0.157482	+	0.987522i	$0.449662\pi$
$224$	0	0
$225$	−15854.1	−0.0208779
$226$	0	0
$227$	−1.10273e6	−1.42038	−0.710191	−	0.704009i	$-0.751391\pi$
$227$	−1.10273e6	−1.42038	−0.710191	+	0.704009i	$0.751391\pi$
$228$	0	0
$229$	−46621.7	−0.0587489	−0.0293744	−	0.999568i	$-0.509352\pi$
$229$	−46621.7	−0.0587489	−0.0293744	+	0.999568i	$0.509352\pi$
$230$	0	0
$231$	614046.	0.757131
$232$	0	0
$233$	995577.	1.20139	0.600696	−	0.799477i	$-0.294890\pi$
$233$	995577.	1.20139	0.600696	+	0.799477i	$0.294890\pi$
$234$	0	0
$235$	1.66812e6	1.97042
$236$	0	0
$237$	292764.	0.338568
$238$	0	0
$239$	−1.40120e6	−1.58674	−0.793372	−	0.608738i	$-0.791676\pi$
$239$	−1.40120e6	−1.58674	−0.793372	+	0.608738i	$0.791676\pi$
$240$	0	0
$241$	−1.51280e6	−1.67780	−0.838899	−	0.544287i	$-0.816800\pi$
$241$	−1.51280e6	−1.67780	−0.838899	+	0.544287i	$0.816800\pi$
$242$	0	0
$243$	−903546.	−0.981601
$244$	0	0
$245$	1.30629e6	1.39035
$246$	0	0
$247$	1.20916e6	1.26107
$248$	0	0
$249$	1.39353e6	1.42436
$250$	0	0
$251$	−39179.9	−0.0392536	−0.0196268	−	0.999807i	$-0.506248\pi$
$251$	−39179.9	−0.0392536	−0.0196268	+	0.999807i	$0.506248\pi$
$252$	0	0
$253$	361426.	0.354992
$254$	0	0
$255$	−1.55624e6	−1.49874
$256$	0	0
$257$	341829.	0.322831	0.161416	−	0.986887i	$-0.448394\pi$
$257$	341829.	0.322831	0.161416	+	0.986887i	$0.448394\pi$
$258$	0	0
$259$	1.89376e6	1.75419
$260$	0	0
$261$	613560.	0.557514
$262$	0	0
$263$	−751539.	−0.669980	−0.334990	−	0.942222i	$-0.608733\pi$
$263$	−751539.	−0.669980	−0.334990	+	0.942222i	$0.608733\pi$
$264$	0	0
$265$	−1.33111e6	−1.16439
$266$	0	0
$267$	−59963.5	−0.0514765
$268$	0	0
$269$	50498.3	0.0425497	0.0212748	−	0.999774i	$-0.493228\pi$
$269$	50498.3	0.0425497	0.0212748	+	0.999774i	$0.493228\pi$
$270$	0	0
$271$	447260.	0.369944	0.184972	−	0.982744i	$-0.440780\pi$
$271$	447260.	0.369944	0.184972	+	0.982744i	$0.440780\pi$
$272$	0	0
$273$	−3.69115e6	−2.99748
$274$	0	0
$275$	−4848.89	−0.00386643
$276$	0	0
$277$	−363369.	−0.284543	−0.142272	−	0.989828i	$-0.545441\pi$
$277$	−363369.	−0.284543	−0.142272	+	0.989828i	$0.545441\pi$
$278$	0	0
$279$	3.76302e6	2.89419
$280$	0	0
$281$	607636.	0.459069	0.229535	−	0.973301i	$-0.426280\pi$
$281$	607636.	0.459069	0.229535	+	0.973301i	$0.426280\pi$
$282$	0	0
$283$	1.08363e6	0.804293	0.402147	−	0.915575i	$-0.368264\pi$
$283$	1.08363e6	0.804293	0.402147	+	0.915575i	$0.368264\pi$
$284$	0	0
$285$	−2.33336e6	−1.70165
$286$	0	0
$287$	1.48025e6	1.06079
$288$	0	0
$289$	−190553.	−0.134206
$290$	0	0
$291$	−200125.	−0.138538
$292$	0	0
$293$	384971.	0.261974	0.130987	−	0.991384i	$-0.458185\pi$
$293$	384971.	0.261974	0.130987	+	0.991384i	$0.458185\pi$
$294$	0	0
$295$	386882.	0.258836
$296$	0	0
$297$	466701.	0.307006
$298$	0	0
$299$	−2.17260e6	−1.40541
$300$	0	0
$301$	1.71265e6	1.08956
$302$	0	0
$303$	1.13773e6	0.711924
$304$	0	0
$305$	−2.72603e6	−1.67796
$306$	0	0
$307$	−76125.6	−0.0460983	−0.0230492	−	0.999734i	$-0.507337\pi$
$307$	−76125.6	−0.0460983	−0.0230492	+	0.999734i	$0.507337\pi$
$308$	0	0
$309$	−2.84616e6	−1.69575
$310$	0	0
$311$	1.25747e6	0.737219	0.368610	−	0.929584i	$-0.379834\pi$
$311$	1.25747e6	0.737219	0.368610	+	0.929584i	$0.379834\pi$
$312$	0	0
$313$	−1.31816e6	−0.760516	−0.380258	−	0.924881i	$-0.624165\pi$
$313$	−1.31816e6	−0.760516	−0.380258	+	0.924881i	$0.624165\pi$
$314$	0	0
$315$	4.41265e6	2.50567
$316$	0	0
$317$	−2.56055e6	−1.43115	−0.715574	−	0.698536i	$-0.753835\pi$
$317$	−2.56055e6	−1.43115	−0.715574	+	0.698536i	$0.753835\pi$
$318$	0	0
$319$	187654.	0.103248
$320$	0	0
$321$	5.23755e6	2.83704
$322$	0	0
$323$	1.84317e6	0.983014
$324$	0	0
$325$	29147.7	0.0153072
$326$	0	0
$327$	−2.40064e6	−1.24153
$328$	0	0
$329$	6.03112e6	3.07191
$330$	0	0
$331$	355258.	0.178227	0.0891135	−	0.996021i	$-0.471597\pi$
$331$	355258.	0.178227	0.0891135	+	0.996021i	$0.471597\pi$
$332$	0	0
$333$	3.73095e6	1.84378
$334$	0	0
$335$	1.32905e6	0.647039
$336$	0	0
$337$	−1.46585e6	−0.703095	−0.351548	−	0.936170i	$-0.614344\pi$
$337$	−1.46585e6	−0.703095	−0.351548	+	0.936170i	$0.614344\pi$
$338$	0	0
$339$	2.13120e6	1.00722
$340$	0	0
$341$	1.15090e6	0.535983
$342$	0	0
$343$	1.34784e6	0.618592
$344$	0	0
$345$	4.19257e6	1.89641
$346$	0	0
$347$	2.14456e6	0.956126	0.478063	−	0.878326i	$-0.341339\pi$
$347$	2.14456e6	0.956126	0.478063	+	0.878326i	$0.341339\pi$
$348$	0	0
$349$	3.12653e6	1.37404	0.687019	−	0.726640i	$-0.258919\pi$
$349$	3.12653e6	1.37404	0.687019	+	0.726640i	$0.258919\pi$
$350$	0	0
$351$	−2.80543e6	−1.21543
$352$	0	0
$353$	368782.	0.157519	0.0787594	−	0.996894i	$-0.474904\pi$
$353$	368782.	0.157519	0.0787594	+	0.996894i	$0.474904\pi$
$354$	0	0
$355$	−104487.	−0.0440040
$356$	0	0
$357$	−5.62659e6	−2.33655
$358$	0	0
$359$	−2.36501e6	−0.968496	−0.484248	−	0.874931i	$-0.660907\pi$
$359$	−2.36501e6	−0.968496	−0.484248	+	0.874931i	$0.660907\pi$
$360$	0	0
$361$	287484.	0.116104
$362$	0	0
$363$	369994.	0.147376
$364$	0	0
$365$	−759515.	−0.298404
$366$	0	0
$367$	1.98689e6	0.770032	0.385016	−	0.922910i	$-0.374196\pi$
$367$	1.98689e6	0.770032	0.385016	+	0.922910i	$0.374196\pi$
$368$	0	0
$369$	2.91628e6	1.11497
$370$	0	0
$371$	−4.81263e6	−1.81530
$372$	0	0
$373$	−100049.	−0.0372339	−0.0186170	−	0.999827i	$-0.505926\pi$
$373$	−100049.	−0.0372339	−0.0186170	+	0.999827i	$0.505926\pi$
$374$	0	0
$375$	−4.44252e6	−1.63137
$376$	0	0
$377$	−1.12802e6	−0.408757
$378$	0	0
$379$	−271345.	−0.0970340	−0.0485170	−	0.998822i	$-0.515450\pi$
$379$	−271345.	−0.0970340	−0.0485170	+	0.998822i	$0.515450\pi$
$380$	0	0
$381$	844481.	0.298042
$382$	0	0
$383$	−3.35065e6	−1.16717	−0.583583	−	0.812054i	$-0.698350\pi$
$383$	−3.35065e6	−1.16717	−0.583583	+	0.812054i	$0.698350\pi$
$384$	0	0
$385$	1.34958e6	0.464032
$386$	0	0
$387$	3.37412e6	1.14521
$388$	0	0
$389$	1.57219e6	0.526782	0.263391	−	0.964689i	$-0.415159\pi$
$389$	1.57219e6	0.526782	0.263391	+	0.964689i	$0.415159\pi$
$390$	0	0
$391$	−3.31180e6	−1.09552
$392$	0	0
$393$	−8.92749e6	−2.91573
$394$	0	0
$395$	643452.	0.207503
$396$	0	0
$397$	2.94505e6	0.937813	0.468907	−	0.883248i	$-0.344648\pi$
$397$	2.94505e6	0.937813	0.468907	+	0.883248i	$0.344648\pi$
$398$	0	0
$399$	−8.43630e6	−2.65289
$400$	0	0
$401$	−480807.	−0.149317	−0.0746586	−	0.997209i	$-0.523787\pi$
$401$	−480807.	−0.149317	−0.0746586	+	0.997209i	$0.523787\pi$
$402$	0	0
$403$	−6.91829e6	−2.12195
$404$	0	0
$405$	74093.3	0.0224461
$406$	0	0
$407$	1.14109e6	0.341455
$408$	0	0
$409$	2.80073e6	0.827872	0.413936	−	0.910306i	$-0.364154\pi$
$409$	2.80073e6	0.827872	0.413936	+	0.910306i	$0.364154\pi$
$410$	0	0
$411$	−4.92895e6	−1.43930
$412$	0	0
$413$	1.39878e6	0.403528
$414$	0	0
$415$	3.06278e6	0.872963
$416$	0	0
$417$	−7.52177e6	−2.11826
$418$	0	0
$419$	4.58561e6	1.27603	0.638017	−	0.770022i	$-0.279755\pi$
$419$	4.58561e6	1.27603	0.638017	+	0.770022i	$0.279755\pi$
$420$	0	0
$421$	−3.65111e6	−1.00397	−0.501984	−	0.864877i	$-0.667396\pi$
$421$	−3.65111e6	−1.00397	−0.501984	+	0.864877i	$0.667396\pi$
$422$	0	0
$423$	1.18820e7	3.22879
$424$	0	0
$425$	44431.0	0.0119320
$426$	0	0
$427$	−9.85598e6	−2.61595
$428$	0	0
$429$	−2.22411e6	−0.583461
$430$	0	0
$431$	3.90343e6	1.01217	0.506085	−	0.862484i	$-0.331092\pi$
$431$	3.90343e6	1.01217	0.506085	+	0.862484i	$0.331092\pi$
$432$	0	0
$433$	−6.51560e6	−1.67007	−0.835035	−	0.550196i	$-0.814553\pi$
$433$	−6.51560e6	−1.67007	−0.835035	+	0.550196i	$0.814553\pi$
$434$	0	0
$435$	2.17680e6	0.551562
$436$	0	0
$437$	−4.96558e6	−1.24385
$438$	0	0
$439$	2.51459e6	0.622738	0.311369	−	0.950289i	$-0.399213\pi$
$439$	2.51459e6	0.622738	0.311369	+	0.950289i	$0.399213\pi$
$440$	0	0
$441$	9.30471e6	2.27828
$442$	0	0
$443$	−1.09010e6	−0.263910	−0.131955	−	0.991256i	$-0.542126\pi$
$443$	−1.09010e6	−0.263910	−0.131955	+	0.991256i	$0.542126\pi$
$444$	0	0
$445$	−131791.	−0.0315490
$446$	0	0
$447$	−833345.	−0.197268
$448$	0	0
$449$	−2.03880e6	−0.477265	−0.238633	−	0.971110i	$-0.576699\pi$
$449$	−2.03880e6	−0.477265	−0.238633	+	0.971110i	$0.576699\pi$
$450$	0	0
$451$	891927.	0.206485
$452$	0	0
$453$	8.23462e6	1.88538
$454$	0	0
$455$	−8.11262e6	−1.83710
$456$	0	0
$457$	−7.74124e6	−1.73388	−0.866942	−	0.498410i	$-0.833918\pi$
$457$	−7.74124e6	−1.73388	−0.866942	+	0.498410i	$0.833918\pi$
$458$	0	0
$459$	−4.27644e6	−0.947438
$460$	0	0
$461$	−5.59266e6	−1.22565	−0.612824	−	0.790219i	$-0.709967\pi$
$461$	−5.59266e6	−1.22565	−0.612824	+	0.790219i	$0.709967\pi$
$462$	0	0
$463$	−796538.	−0.172685	−0.0863423	−	0.996266i	$-0.527518\pi$
$463$	−796538.	−0.172685	−0.0863423	+	0.996266i	$0.527518\pi$
$464$	0	0
$465$	1.33505e7	2.86329
$466$	0	0
$467$	−1.88198e6	−0.399322	−0.199661	−	0.979865i	$-0.563984\pi$
$467$	−1.88198e6	−0.399322	−0.199661	+	0.979865i	$0.563984\pi$
$468$	0	0
$469$	4.80521e6	1.00874
$470$	0	0
$471$	9.17309e6	1.90530
$472$	0	0
$473$	1.03196e6	0.212084
$474$	0	0
$475$	66618.3	0.0135475
$476$	0	0
$477$	−9.48147e6	−1.90801
$478$	0	0
$479$	−7.39299e6	−1.47225	−0.736125	−	0.676846i	$-0.763346\pi$
$479$	−7.39299e6	−1.47225	−0.736125	+	0.676846i	$0.763346\pi$
$480$	0	0
$481$	−6.85931e6	−1.35182
$482$	0	0
$483$	1.51583e7	2.95653
$484$	0	0
$485$	−439845.	−0.0849074
$486$	0	0
$487$	−2.44572e6	−0.467288	−0.233644	−	0.972322i	$-0.575065\pi$
$487$	−2.44572e6	−0.467288	−0.233644	+	0.972322i	$0.575065\pi$
$488$	0	0
$489$	−1.51044e7	−2.85648
$490$	0	0
$491$	−555739.	−0.104032	−0.0520160	−	0.998646i	$-0.516565\pi$
$491$	−555739.	−0.104032	−0.0520160	+	0.998646i	$0.516565\pi$
$492$	0	0
$493$	−1.71950e6	−0.318628
$494$	0	0
$495$	2.65884e6	0.487731
$496$	0	0
$497$	−377775.	−0.0686028
$498$	0	0
$499$	−8.28233e6	−1.48902	−0.744511	−	0.667610i	$-0.767317\pi$
$499$	−8.28233e6	−1.48902	−0.744511	+	0.667610i	$0.767317\pi$
$500$	0	0
$501$	−1.19889e7	−2.13396
$502$	0	0
$503$	2.15819e6	0.380337	0.190169	−	0.981751i	$-0.439097\pi$
$503$	2.15819e6	0.380337	0.190169	+	0.981751i	$0.439097\pi$
$504$	0	0
$505$	2.50057e6	0.436325
$506$	0	0
$507$	3.98658e6	0.688780
$508$	0	0
$509$	7.95956e6	1.36174	0.680870	−	0.732404i	$-0.261602\pi$
$509$	7.95956e6	1.36174	0.680870	+	0.732404i	$0.261602\pi$
$510$	0	0
$511$	−2.74603e6	−0.465215
$512$	0	0
$513$	−6.41194e6	−1.07571
$514$	0	0
$515$	−6.25544e6	−1.03930
$516$	0	0
$517$	3.63405e6	0.597950
$518$	0	0
$519$	−1.23140e6	−0.200670
$520$	0	0
$521$	−3.85570e6	−0.622312	−0.311156	−	0.950359i	$-0.600716\pi$
$521$	−3.85570e6	−0.622312	−0.311156	+	0.950359i	$0.600716\pi$
$522$	0	0
$523$	−7.82125e6	−1.25032	−0.625161	−	0.780496i	$-0.714967\pi$
$523$	−7.82125e6	−1.25032	−0.625161	+	0.780496i	$0.714967\pi$
$524$	0	0
$525$	−203363.	−0.0322014
$526$	0	0
$527$	−1.05458e7	−1.65407
$528$	0	0
$529$	2.48578e6	0.386210
$530$	0	0
$531$	2.75576e6	0.424136
$532$	0	0
$533$	−5.36155e6	−0.817471
$534$	0	0
$535$	1.15114e7	1.73877
$536$	0	0
$537$	−1.55435e7	−2.32601
$538$	0	0
$539$	2.84579e6	0.421921
$540$	0	0
$541$	−1.17121e6	−0.172044	−0.0860221	−	0.996293i	$-0.527416\pi$
$541$	−1.17121e6	−0.172044	−0.0860221	+	0.996293i	$0.527416\pi$
$542$	0	0
$543$	−2.56599e6	−0.373469
$544$	0	0
$545$	−5.27627e6	−0.760914
$546$	0	0
$547$	−4.98116e6	−0.711807	−0.355904	−	0.934523i	$-0.615827\pi$
$547$	−4.98116e6	−0.711807	−0.355904	+	0.934523i	$0.615827\pi$
$548$	0	0
$549$	−1.94175e7	−2.74955
$550$	0	0
$551$	−2.57815e6	−0.361767
$552$	0	0
$553$	2.32641e6	0.323499
$554$	0	0
$555$	1.32367e7	1.82409
$556$	0	0
$557$	7.59988e6	1.03793	0.518966	−	0.854795i	$-0.326317\pi$
$557$	7.59988e6	1.03793	0.518966	+	0.854795i	$0.326317\pi$
$558$	0	0
$559$	−6.20329e6	−0.839639
$560$	0	0
$561$	−3.39030e6	−0.454811
$562$	0	0
$563$	−4.85731e6	−0.645840	−0.322920	−	0.946426i	$-0.604664\pi$
$563$	−4.85731e6	−0.645840	−0.322920	+	0.946426i	$0.604664\pi$
$564$	0	0
$565$	4.68406e6	0.617308
$566$	0	0
$567$	267885.	0.0349938
$568$	0	0
$569$	4.86673e6	0.630169	0.315084	−	0.949064i	$-0.397967\pi$
$569$	4.86673e6	0.630169	0.315084	+	0.949064i	$0.397967\pi$
$570$	0	0
$571$	−647211.	−0.0830722	−0.0415361	−	0.999137i	$-0.513225\pi$
$571$	−647211.	−0.0830722	−0.0415361	+	0.999137i	$0.513225\pi$
$572$	0	0
$573$	1.04399e6	0.132834
$574$	0	0
$575$	−119699.	−0.0150981
$576$	0	0
$577$	6.28194e6	0.785515	0.392757	−	0.919642i	$-0.371521\pi$
$577$	6.28194e6	0.785515	0.392757	+	0.919642i	$0.371521\pi$
$578$	0	0
$579$	−1.23202e7	−1.52729
$580$	0	0
$581$	1.10735e7	1.36096
$582$	0	0
$583$	−2.89985e6	−0.353349
$584$	0	0
$585$	−1.59828e7	−1.93092
$586$	0	0
$587$	1.43029e7	1.71328	0.856638	−	0.515917i	$-0.172549\pi$
$587$	1.43029e7	1.71328	0.856638	+	0.515917i	$0.172549\pi$
$588$	0	0
$589$	−1.58120e7	−1.87802
$590$	0	0
$591$	9.94604e6	1.17134
$592$	0	0
$593$	1.11657e7	1.30392	0.651960	−	0.758254i	$-0.273947\pi$
$593$	1.11657e7	1.30392	0.651960	+	0.758254i	$0.273947\pi$
$594$	0	0
$595$	−1.23664e7	−1.43203
$596$	0	0
$597$	−7.45968e6	−0.856612
$598$	0	0
$599$	−2.22282e6	−0.253126	−0.126563	−	0.991959i	$-0.540395\pi$
$599$	−2.22282e6	−0.253126	−0.126563	+	0.991959i	$0.540395\pi$
$600$	0	0
$601$	1.25195e7	1.41384	0.706920	−	0.707294i	$-0.250084\pi$
$601$	1.25195e7	1.41384	0.706920	+	0.707294i	$0.250084\pi$
$602$	0	0
$603$	9.46684e6	1.06026
$604$	0	0
$605$	813192.	0.0903243
$606$	0	0
$607$	9.80774e6	1.08043	0.540216	−	0.841527i	$-0.318343\pi$
$607$	9.80774e6	1.08043	0.540216	+	0.841527i	$0.318343\pi$
$608$	0	0
$609$	7.87023e6	0.859893
$610$	0	0
$611$	−2.18450e7	−2.36728
$612$	0	0
$613$	1.04278e7	1.12084	0.560418	−	0.828210i	$-0.310641\pi$
$613$	1.04278e7	1.12084	0.560418	+	0.828210i	$0.310641\pi$
$614$	0	0
$615$	1.03464e7	1.10307
$616$	0	0
$617$	−1.83918e6	−0.194497	−0.0972483	−	0.995260i	$-0.531004\pi$
$617$	−1.83918e6	−0.194497	−0.0972483	+	0.995260i	$0.531004\pi$
$618$	0	0
$619$	−185199.	−0.0194272	−0.00971362	−	0.999953i	$-0.503092\pi$
$619$	−185199.	−0.0194272	−0.00971362	+	0.999953i	$0.503092\pi$
$620$	0	0
$621$	1.15209e7	1.19883
$622$	0	0
$623$	−476492.	−0.0491853
$624$	0	0
$625$	−9.63879e6	−0.987012
$626$	0	0
$627$	−5.08329e6	−0.516388
$628$	0	0
$629$	−1.04559e7	−1.05375
$630$	0	0
$631$	−6.47412e6	−0.647303	−0.323651	−	0.946176i	$-0.604910\pi$
$631$	−6.47412e6	−0.647303	−0.323651	+	0.946176i	$0.604910\pi$
$632$	0	0
$633$	1.21800e7	1.20820
$634$	0	0
$635$	1.85605e6	0.182665
$636$	0	0
$637$	−1.71066e7	−1.67038
$638$	0	0
$639$	−744262.	−0.0721064
$640$	0	0
$641$	4.64672e6	0.446685	0.223343	−	0.974740i	$-0.428303\pi$
$641$	4.64672e6	0.446685	0.223343	+	0.974740i	$0.428303\pi$
$642$	0	0
$643$	1.28549e7	1.22614	0.613072	−	0.790027i	$-0.289933\pi$
$643$	1.28549e7	1.22614	0.613072	+	0.790027i	$0.289933\pi$
$644$	0	0
$645$	1.19708e7	1.13298
$646$	0	0
$647$	1.61104e7	1.51302	0.756510	−	0.653982i	$-0.226903\pi$
$647$	1.61104e7	1.51302	0.756510	+	0.653982i	$0.226903\pi$
$648$	0	0
$649$	842834.	0.0785471
$650$	0	0
$651$	4.82689e7	4.46391
$652$	0	0
$653$	−4.72189e6	−0.433344	−0.216672	−	0.976244i	$-0.569520\pi$
$653$	−4.72189e6	−0.433344	−0.216672	+	0.976244i	$0.569520\pi$
$654$	0	0
$655$	−1.96213e7	−1.78700
$656$	0	0
$657$	−5.41002e6	−0.488974
$658$	0	0
$659$	6.08366e6	0.545697	0.272848	−	0.962057i	$-0.412034\pi$
$659$	6.08366e6	0.545697	0.272848	+	0.962057i	$0.412034\pi$
$660$	0	0
$661$	−1.01649e6	−0.0904897	−0.0452448	−	0.998976i	$-0.514407\pi$
$661$	−1.01649e6	−0.0904897	−0.0452448	+	0.998976i	$0.514407\pi$
$662$	0	0
$663$	2.03798e7	1.80059
$664$	0	0
$665$	−1.85418e7	−1.62591
$666$	0	0
$667$	4.63240e6	0.403173
$668$	0	0
$669$	5.91080e6	0.510600
$670$	0	0
$671$	−5.93872e6	−0.509198
$672$	0	0
$673$	1.00418e7	0.854619	0.427310	−	0.904105i	$-0.359461\pi$
$673$	1.00418e7	0.854619	0.427310	+	0.904105i	$0.359461\pi$
$674$	0	0
$675$	−154565.	−0.0130572
$676$	0	0
$677$	−1.12734e7	−0.945331	−0.472665	−	0.881242i	$-0.656708\pi$
$677$	−1.12734e7	−0.945331	−0.472665	+	0.881242i	$0.656708\pi$
$678$	0	0
$679$	−1.59027e6	−0.132372
$680$	0	0
$681$	−2.78672e7	−2.30264
$682$	0	0
$683$	2.74328e6	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$683$	2.74328e6	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$684$	0	0
$685$	−1.08331e7	−0.882119
$686$	0	0
$687$	−1.17818e6	−0.0952401
$688$	0	0
$689$	1.74316e7	1.39891
$690$	0	0
$691$	1.31710e7	1.04936	0.524680	−	0.851300i	$-0.324185\pi$
$691$	1.31710e7	1.04936	0.524680	+	0.851300i	$0.324185\pi$
$692$	0	0
$693$	9.61308e6	0.760378
$694$	0	0
$695$	−1.65318e7	−1.29825
$696$	0	0
$697$	−8.17285e6	−0.637223
$698$	0	0
$699$	2.51593e7	1.94763
$700$	0	0
$701$	−1.76568e7	−1.35712	−0.678558	−	0.734546i	$-0.737395\pi$
$701$	−1.76568e7	−1.35712	−0.678558	+	0.734546i	$0.737395\pi$
$702$	0	0
$703$	−1.56773e7	−1.19641
$704$	0	0
$705$	4.21552e7	3.19432
$706$	0	0
$707$	9.04084e6	0.680237
$708$	0	0
$709$	1.58423e7	1.18360	0.591798	−	0.806086i	$-0.298418\pi$
$709$	1.58423e7	1.18360	0.591798	+	0.806086i	$0.298418\pi$
$710$	0	0
$711$	4.58331e6	0.340020
$712$	0	0
$713$	2.84110e7	2.09297
$714$	0	0
$715$	−4.88826e6	−0.357593
$716$	0	0
$717$	−3.54099e7	−2.57233
$718$	0	0
$719$	1.10799e7	0.799305	0.399652	−	0.916667i	$-0.369131\pi$
$719$	1.10799e7	0.799305	0.399652	+	0.916667i	$0.369131\pi$
$720$	0	0
$721$	−2.26166e7	−1.62028
$722$	0	0
$723$	−3.82301e7	−2.71994
$724$	0	0
$725$	−62148.2	−0.00439121
$726$	0	0
$727$	−9.71158e6	−0.681481	−0.340741	−	0.940157i	$-0.610678\pi$
$727$	−9.71158e6	−0.681481	−0.340741	+	0.940157i	$0.610678\pi$
$728$	0	0
$729$	−2.31577e7	−1.61390
$730$	0	0
$731$	−9.45595e6	−0.654503
$732$	0	0
$733$	−2.45714e6	−0.168916	−0.0844580	−	0.996427i	$-0.526916\pi$
$733$	−2.45714e6	−0.168916	−0.0844580	+	0.996427i	$0.526916\pi$
$734$	0	0
$735$	3.30114e7	2.25395
$736$	0	0
$737$	2.89538e6	0.196353
$738$	0	0
$739$	−2.22772e6	−0.150055	−0.0750273	−	0.997181i	$-0.523904\pi$
$739$	−2.22772e6	−0.150055	−0.0750273	+	0.997181i	$0.523904\pi$
$740$	0	0
$741$	3.05567e7	2.04438
$742$	0	0
$743$	1.01867e7	0.676960	0.338480	−	0.940974i	$-0.390087\pi$
$743$	1.01867e7	0.676960	0.338480	+	0.940974i	$0.390087\pi$
$744$	0	0
$745$	−1.83157e6	−0.120902
$746$	0	0
$747$	2.18162e7	1.43046
$748$	0	0
$749$	4.16195e7	2.71077
$750$	0	0
$751$	6.45437e6	0.417594	0.208797	−	0.977959i	$-0.433045\pi$
$751$	6.45437e6	0.417594	0.208797	+	0.977959i	$0.433045\pi$
$752$	0	0
$753$	−990119.	−0.0636355
$754$	0	0
$755$	1.80985e7	1.15551
$756$	0	0
$757$	1.46247e7	0.927571	0.463785	−	0.885948i	$-0.346491\pi$
$757$	1.46247e7	0.927571	0.463785	+	0.885948i	$0.346491\pi$
$758$	0	0
$759$	9.13362e6	0.575491
$760$	0	0
$761$	−2.73756e7	−1.71357	−0.856784	−	0.515675i	$-0.827541\pi$
$761$	−2.73756e7	−1.71357	−0.856784	+	0.515675i	$0.827541\pi$
$762$	0	0
$763$	−1.90764e7	−1.18627
$764$	0	0
$765$	−2.43634e7	−1.50516
$766$	0	0
$767$	−5.06644e6	−0.310967
$768$	0	0
$769$	−1.18927e7	−0.725213	−0.362607	−	0.931942i	$-0.618113\pi$
$769$	−1.18927e7	−0.725213	−0.362607	+	0.931942i	$0.618113\pi$
$770$	0	0
$771$	8.63837e6	0.523354
$772$	0	0
$773$	1.16963e7	0.704042	0.352021	−	0.935992i	$-0.385495\pi$
$773$	1.16963e7	0.704042	0.352021	+	0.935992i	$0.385495\pi$
$774$	0	0
$775$	−381161.	−0.0227958
$776$	0	0
$777$	4.78574e7	2.84379
$778$	0	0
$779$	−1.22541e7	−0.723496
$780$	0	0
$781$	−227628.	−0.0133536
$782$	0	0
$783$	5.98170e6	0.348675
$784$	0	0
$785$	2.01611e7	1.16772
$786$	0	0
$787$	−2.48828e6	−0.143206	−0.0716031	−	0.997433i	$-0.522811\pi$
$787$	−2.48828e6	−0.143206	−0.0716031	+	0.997433i	$0.522811\pi$
$788$	0	0
$789$	−1.89922e7	−1.08613
$790$	0	0
$791$	1.69353e7	0.962390
$792$	0	0
$793$	3.56988e7	2.01591
$794$	0	0
$795$	−3.36385e7	−1.88764
$796$	0	0
$797$	−2.98889e7	−1.66673	−0.833363	−	0.552726i	$-0.813588\pi$
$797$	−2.98889e7	−1.66673	−0.833363	+	0.552726i	$0.813588\pi$
$798$	0	0
$799$	−3.32993e7	−1.84531
$800$	0	0
$801$	−938747.	−0.0516972
$802$	0	0
$803$	−1.65462e6	−0.0905546
$804$	0	0
$805$	3.33157e7	1.81200
$806$	0	0
$807$	1.27615e6	0.0689789
$808$	0	0
$809$	−2.61207e7	−1.40318	−0.701590	−	0.712581i	$-0.747526\pi$
$809$	−2.61207e7	−1.40318	−0.701590	+	0.712581i	$0.747526\pi$
$810$	0	0
$811$	−2.19410e7	−1.17140	−0.585699	−	0.810529i	$-0.699180\pi$
$811$	−2.19410e7	−1.17140	−0.585699	+	0.810529i	$0.699180\pi$
$812$	0	0
$813$	1.13027e7	0.599731
$814$	0	0
$815$	−3.31973e7	−1.75069
$816$	0	0
$817$	−1.41779e7	−0.743116
$818$	0	0
$819$	−5.77862e7	−3.01033
$820$	0	0
$821$	−4.75760e6	−0.246337	−0.123169	−	0.992386i	$-0.539306\pi$
$821$	−4.75760e6	−0.246337	−0.123169	+	0.992386i	$0.539306\pi$
$822$	0	0
$823$	6.19431e6	0.318782	0.159391	−	0.987216i	$-0.449047\pi$
$823$	6.19431e6	0.318782	0.159391	+	0.987216i	$0.449047\pi$
$824$	0	0
$825$	−122537.	−0.00626803
$826$	0	0
$827$	1.41552e7	0.719699	0.359850	−	0.933010i	$-0.382828\pi$
$827$	1.41552e7	0.719699	0.359850	+	0.933010i	$0.382828\pi$
$828$	0	0
$829$	1.19032e7	0.601556	0.300778	−	0.953694i	$-0.402754\pi$
$829$	1.19032e7	0.601556	0.300778	+	0.953694i	$0.402754\pi$
$830$	0	0
$831$	−9.18271e6	−0.461284
$832$	0	0
$833$	−2.60764e7	−1.30207
$834$	0	0
$835$	−2.63499e7	−1.30786
$836$	0	0
$837$	3.66864e7	1.81005
$838$	0	0
$839$	3.80619e7	1.86675	0.933374	−	0.358904i	$-0.116850\pi$
$839$	3.80619e7	1.86675	0.933374	+	0.358904i	$0.116850\pi$
$840$	0	0
$841$	−1.81060e7	−0.882739
$842$	0	0
$843$	1.53556e7	0.744215
$844$	0	0
$845$	8.76192e6	0.422141
$846$	0	0
$847$	2.94011e6	0.140817
$848$	0	0
$849$	2.73844e7	1.30387
$850$	0	0
$851$	2.81688e7	1.33335
$852$	0	0
$853$	−8.91252e6	−0.419400	−0.209700	−	0.977766i	$-0.567249\pi$
$853$	−8.91252e6	−0.419400	−0.209700	+	0.977766i	$0.567249\pi$
$854$	0	0
$855$	−3.65295e7	−1.70895
$856$	0	0
$857$	2.38385e7	1.10873	0.554366	−	0.832273i	$-0.312961\pi$
$857$	2.38385e7	1.10873	0.554366	+	0.832273i	$0.312961\pi$
$858$	0	0
$859$	7.15819e6	0.330994	0.165497	−	0.986210i	$-0.447077\pi$
$859$	7.15819e6	0.330994	0.165497	+	0.986210i	$0.447077\pi$
$860$	0	0
$861$	3.74076e7	1.71970
$862$	0	0
$863$	2.33903e7	1.06908	0.534539	−	0.845144i	$-0.320485\pi$
$863$	2.33903e7	1.06908	0.534539	+	0.845144i	$0.320485\pi$
$864$	0	0
$865$	−2.70645e6	−0.122987
$866$	0	0
$867$	−4.81548e6	−0.217567
$868$	0	0
$869$	1.40178e6	0.0629694
$870$	0	0
$871$	−1.74047e7	−0.777358
$872$	0	0
$873$	−3.13302e6	−0.139132
$874$	0	0
$875$	−3.53019e7	−1.55876
$876$	0	0
$877$	4.19751e7	1.84286	0.921431	−	0.388543i	$-0.127021\pi$
$877$	4.19751e7	1.84286	0.921431	+	0.388543i	$0.127021\pi$
$878$	0	0
$879$	9.72862e6	0.424697
$880$	0	0
$881$	−7.55945e6	−0.328134	−0.164067	−	0.986449i	$-0.552461\pi$
$881$	−7.55945e6	−0.328134	−0.164067	+	0.986449i	$0.552461\pi$
$882$	0	0
$883$	2.89794e7	1.25080	0.625400	−	0.780304i	$-0.284936\pi$
$883$	2.89794e7	1.25080	0.625400	+	0.780304i	$0.284936\pi$
$884$	0	0
$885$	9.77693e6	0.419608
$886$	0	0
$887$	−4.19301e7	−1.78944	−0.894719	−	0.446629i	$-0.852624\pi$
$887$	−4.19301e7	−1.78944	−0.894719	+	0.446629i	$0.852624\pi$
$888$	0	0
$889$	6.71056e6	0.284776
$890$	0	0
$891$	161414.	0.00681158
$892$	0	0
$893$	−4.99278e7	−2.09514
$894$	0	0
$895$	−3.41623e7	−1.42557
$896$	0	0
$897$	−5.49040e7	−2.27836
$898$	0	0
$899$	1.47511e7	0.608730
$900$	0	0
$901$	2.65717e7	1.09046
$902$	0	0
$903$	4.32804e7	1.76633
$904$	0	0
$905$	−5.63966e6	−0.228893
$906$	0	0
$907$	7.79438e6	0.314603	0.157302	−	0.987551i	$-0.449721\pi$
$907$	7.79438e6	0.314603	0.157302	+	0.987551i	$0.449721\pi$
$908$	0	0
$909$	1.78116e7	0.714977
$910$	0	0
$911$	−2.09270e7	−0.835431	−0.417715	−	0.908578i	$-0.637169\pi$
$911$	−2.09270e7	−0.835431	−0.417715	+	0.908578i	$0.637169\pi$
$912$	0	0
$913$	6.67235e6	0.264912
$914$	0	0
$915$	−6.88895e7	−2.72020
$916$	0	0
$917$	−7.09411e7	−2.78596
$918$	0	0
$919$	6.21354e6	0.242689	0.121344	−	0.992610i	$-0.461279\pi$
$919$	6.21354e6	0.242689	0.121344	+	0.992610i	$0.461279\pi$
$920$	0	0
$921$	−1.92378e6	−0.0747318
$922$	0	0
$923$	1.36832e6	0.0528668
$924$	0	0
$925$	−377912.	−0.0145223
$926$	0	0
$927$	−4.45575e7	−1.70303
$928$	0	0
$929$	1.86038e7	0.707234	0.353617	−	0.935390i	$-0.384952\pi$
$929$	1.86038e7	0.707234	0.353617	+	0.935390i	$0.384952\pi$
$930$	0	0
$931$	−3.90979e7	−1.47836
$932$	0	0
$933$	3.17776e7	1.19514
$934$	0	0
$935$	−7.45139e6	−0.278746
$936$	0	0
$937$	5.27518e7	1.96286	0.981428	−	0.191832i	$-0.0614429\pi$
$937$	5.27518e7	1.96286	0.981428	+	0.191832i	$0.0614429\pi$
$938$	0	0
$939$	−3.33114e7	−1.23290
$940$	0	0
$941$	−3.67919e7	−1.35450	−0.677249	−	0.735754i	$-0.736828\pi$
$941$	−3.67919e7	−1.35450	−0.677249	+	0.735754i	$0.736828\pi$
$942$	0	0
$943$	2.20180e7	0.806304
$944$	0	0
$945$	4.30197e7	1.56707
$946$	0	0
$947$	−3.70930e7	−1.34406	−0.672028	−	0.740526i	$-0.734576\pi$
$947$	−3.70930e7	−1.34406	−0.672028	+	0.740526i	$0.734576\pi$
$948$	0	0
$949$	9.94627e6	0.358505
$950$	0	0
$951$	−6.47078e7	−2.32009
$952$	0	0
$953$	−1.77859e7	−0.634373	−0.317187	−	0.948363i	$-0.602738\pi$
$953$	−1.77859e7	−0.634373	−0.317187	+	0.948363i	$0.602738\pi$
$954$	0	0
$955$	2.29454e6	0.0814119
$956$	0	0
$957$	4.74221e6	0.167379
$958$	0	0
$959$	−3.91673e7	−1.37523
$960$	0	0
$961$	6.18407e7	2.16006
$962$	0	0
$963$	8.19955e7	2.84921
$964$	0	0
$965$	−2.70781e7	−0.936051
$966$	0	0
$967$	−1.30531e7	−0.448897	−0.224448	−	0.974486i	$-0.572058\pi$
$967$	−1.30531e7	−0.448897	−0.224448	+	0.974486i	$0.572058\pi$
$968$	0	0
$969$	4.65789e7	1.59360
$970$	0	0
$971$	−1.33944e7	−0.455908	−0.227954	−	0.973672i	$-0.573204\pi$
$971$	−1.33944e7	−0.455908	−0.227954	+	0.973672i	$0.573204\pi$
$972$	0	0
$973$	−5.97708e7	−2.02398
$974$	0	0
$975$	736592.	0.0248151
$976$	0	0
$977$	−5.84912e7	−1.96044	−0.980221	−	0.197905i	$-0.936586\pi$
$977$	−5.84912e7	−1.96044	−0.980221	+	0.197905i	$0.936586\pi$
$978$	0	0
$979$	−287110.	−0.00957397
$980$	0	0
$981$	−3.75828e7	−1.24686
$982$	0	0
$983$	2.61534e7	0.863265	0.431632	−	0.902050i	$-0.357938\pi$
$983$	2.61534e7	0.863265	0.431632	+	0.902050i	$0.357938\pi$
$984$	0	0
$985$	2.18600e7	0.717891
$986$	0	0
$987$	1.52413e8	4.97999
$988$	0	0
$989$	2.54747e7	0.828169
$990$	0	0
$991$	5.29034e7	1.71119	0.855597	−	0.517643i	$-0.173190\pi$
$991$	5.29034e7	1.71119	0.855597	+	0.517643i	$0.173190\pi$
$992$	0	0
$993$	8.97774e6	0.288931
$994$	0	0
$995$	−1.63953e7	−0.525002
$996$	0	0
$997$	4.23746e7	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$997$	4.23746e7	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$998$	0	0
$999$	3.63736e7	1.15312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.6.a.n.1.2		2
4.3	odd	2		704.6.a.m.1.1		2
8.3	odd	2		176.6.a.g.1.2		2
8.5	even	2		44.6.a.b.1.1	✓	2
24.5	odd	2		396.6.a.f.1.2		2
40.13	odd	4		1100.6.b.c.749.1		4
40.29	even	2		1100.6.a.b.1.2		2
40.37	odd	4		1100.6.b.c.749.4		4
88.21	odd	2		484.6.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
44.6.a.b.1.1	✓	2	8.5	even	2
176.6.a.g.1.2		2	8.3	odd	2
396.6.a.f.1.2		2	24.5	odd	2
484.6.a.d.1.1		2	88.21	odd	2
704.6.a.m.1.1		2	4.3	odd	2
704.6.a.n.1.2		2	1.1	even	1	trivial
1100.6.a.b.1.2		2	40.29	even	2
1100.6.b.c.749.1		4	40.13	odd	4
1100.6.b.c.749.4		4	40.37	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:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)

\(f(q)\)	\(=\)	\(q+25.2711 q^{3} +55.5421 q^{5} +200.813 q^{7} +395.626 q^{9} +121.000 q^{11} -727.355 q^{13} +1403.61 q^{15} -1108.74 q^{17} -1662.40 q^{19} +5074.76 q^{21} +2986.99 q^{23} -40.0735 q^{25} +3857.03 q^{27} +1550.86 q^{29} +9511.56 q^{31} +3057.80 q^{33} +11153.6 q^{35} +9430.48 q^{37} -18381.0 q^{39} +7371.29 q^{41} +8528.56 q^{43} +21973.9 q^{45} +30033.5 q^{47} +23518.9 q^{49} -28019.0 q^{51} -23965.7 q^{53} +6720.60 q^{55} -42010.7 q^{57} +6965.57 q^{59} -49080.3 q^{61} +79447.0 q^{63} -40398.9 q^{65} +23928.7 q^{67} +75484.5 q^{69} -1881.22 q^{71} -13674.6 q^{73} -1012.70 q^{75} +24298.4 q^{77} +11584.9 q^{79} +1334.00 q^{81} +55143.4 q^{83} -61581.7 q^{85} +39191.8 q^{87} -2372.81 q^{89} -146063. q^{91} +240367. q^{93} -92333.4 q^{95} -7919.13 q^{97} +47870.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 22 q^{5} + 268 q^{7} + 524 q^{9} + 242 q^{11} - 1232 q^{13} + 2050 q^{15} - 124 q^{17} - 1944 q^{19} + 3780 q^{21} + 3346 q^{23} - 2040 q^{25} + 6066 q^{27} + 6576 q^{29} + 2498 q^{31} + 726 q^{33}+ \cdots + 63404 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 22 * q^5 + 268 * q^7 + 524 * q^9 + 242 * q^11 - 1232 * q^13 + 2050 * q^15 - 124 * q^17 - 1944 * q^19 + 3780 * q^21 + 3346 * q^23 - 2040 * q^25 + 6066 * q^27 + 6576 * q^29 + 2498 * q^31 + 726 * q^33 + 8900 * q^35 + 14674 * q^37 - 8656 * q^39 - 6504 * q^41 - 11628 * q^43 + 17668 * q^45 + 36816 * q^47 + 11226 * q^49 - 46996 * q^51 + 3292 * q^53 + 2662 * q^55 - 36584 * q^57 - 12126 * q^59 - 73128 * q^61 + 88072 * q^63 - 23472 * q^65 - 29334 * q^67 + 68566 * q^69 - 46122 * q^71 + 8240 * q^73 + 37528 * q^75 + 32428 * q^77 - 14780 * q^79 - 72430 * q^81 + 74564 * q^83 - 94612 * q^85 - 57648 * q^87 - 33698 * q^89 - 179968 * q^91 + 375526 * q^93 - 82888 * q^95 + 126162 * q^97 + 63404 * q^99

Introduction

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