LMFDB - Embedded newform 1104.6.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.42196$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +99.7124 q^{5} -125.983 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +99.7124 q^{5} -125.983 q^{7} +81.0000 q^{9} -177.715 q^{11} -919.618 q^{13} -897.412 q^{15} +1568.24 q^{17} -447.960 q^{19} +1133.85 q^{21} +529.000 q^{23} +6817.57 q^{25} -729.000 q^{27} -1298.72 q^{29} +6651.88 q^{31} +1599.43 q^{33} -12562.1 q^{35} -8056.80 q^{37} +8276.56 q^{39} +17941.4 q^{41} -10710.7 q^{43} +8076.71 q^{45} +17886.9 q^{47} -935.293 q^{49} -14114.1 q^{51} -20343.9 q^{53} -17720.4 q^{55} +4031.64 q^{57} -28488.5 q^{59} +16222.3 q^{61} -10204.6 q^{63} -91697.4 q^{65} -54128.0 q^{67} -4761.00 q^{69} -42505.6 q^{71} +40936.2 q^{73} -61358.1 q^{75} +22389.1 q^{77} +24760.6 q^{79} +6561.00 q^{81} +30825.2 q^{83} +156373. q^{85} +11688.4 q^{87} -66531.6 q^{89} +115856. q^{91} -59866.9 q^{93} -44667.2 q^{95} -104506. q^{97} -14394.9 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) $ 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9	$ 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) $ 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9 - 1100 * q^11 - 978 * q^13 - 846 * q^15 + 2522 * q^17 - 2060 * q^19 + 2448 * q^21 + 2645 * q^23 + 12035 * q^25 - 3645 * q^27 + 1526 * q^29 + 7392 * q^31 + 9900 * q^33 - 6056 * q^35 - 8210 * q^37 + 8802 * q^39 + 21250 * q^41 + 4548 * q^43 + 7614 * q^45 - 536 * q^47 - 27979 * q^49 - 22698 * q^51 - 11482 * q^53 + 77064 * q^55 + 18540 * q^57 - 74676 * q^59 - 44618 * q^61 - 22032 * q^63 - 24388 * q^65 + 1412 * q^67 - 23805 * q^69 - 37912 * q^71 + 46546 * q^73 - 108315 * q^75 + 157008 * q^77 - 50544 * q^79 + 32805 * q^81 - 89588 * q^83 + 147892 * q^85 - 13734 * q^87 + 280410 * q^89 + 27416 * q^91 - 66528 * q^93 - 203120 * q^95 + 90074 * q^97 - 89100 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	99.7124	1.78371	0.891855	−	0.452321i	$-0.149404\pi$
$5$	99.7124	1.78371	0.891855	+	0.452321i	$0.149404\pi$
$6$	0	0
$7$	−125.983	−0.971777	−0.485889	−	0.874021i	$-0.661504\pi$
$7$	−125.983	−0.971777	−0.485889	+	0.874021i	$0.661504\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−177.715	−0.442835	−0.221418	−	0.975179i	$-0.571068\pi$
$11$	−177.715	−0.442835	−0.221418	+	0.975179i	$0.571068\pi$
$12$	0	0
$13$	−919.618	−1.50921	−0.754604	−	0.656180i	$-0.772171\pi$
$13$	−919.618	−1.50921	−0.754604	+	0.656180i	$0.772171\pi$
$14$	0	0
$15$	−897.412	−1.02983
$16$	0	0
$17$	1568.24	1.31610	0.658051	−	0.752973i	$-0.271381\pi$
$17$	1568.24	1.31610	0.658051	+	0.752973i	$0.271381\pi$
$18$	0	0
$19$	−447.960	−0.284679	−0.142339	−	0.989818i	$-0.545462\pi$
$19$	−447.960	−0.284679	−0.142339	+	0.989818i	$0.545462\pi$
$20$	0	0
$21$	1133.85	0.561056
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	6817.57	2.18162
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−1298.72	−0.286760	−0.143380	−	0.989668i	$-0.545797\pi$
$29$	−1298.72	−0.286760	−0.143380	+	0.989668i	$0.545797\pi$
$30$	0	0
$31$	6651.88	1.24320	0.621599	−	0.783336i	$-0.286483\pi$
$31$	6651.88	1.24320	0.621599	+	0.783336i	$0.286483\pi$
$32$	0	0
$33$	1599.43	0.255671
$34$	0	0
$35$	−12562.1	−1.73337
$36$	0	0
$37$	−8056.80	−0.967516	−0.483758	−	0.875202i	$-0.660728\pi$
$37$	−8056.80	−0.967516	−0.483758	+	0.875202i	$0.660728\pi$
$38$	0	0
$39$	8276.56	0.871342
$40$	0	0
$41$	17941.4	1.66685	0.833427	−	0.552630i	$-0.186376\pi$
$41$	17941.4	1.66685	0.833427	+	0.552630i	$0.186376\pi$
$42$	0	0
$43$	−10710.7	−0.883377	−0.441688	−	0.897169i	$-0.645620\pi$
$43$	−10710.7	−0.883377	−0.441688	+	0.897169i	$0.645620\pi$
$44$	0	0
$45$	8076.71	0.594570
$46$	0	0
$47$	17886.9	1.18111	0.590555	−	0.806997i	$-0.298909\pi$
$47$	17886.9	1.18111	0.590555	+	0.806997i	$0.298909\pi$
$48$	0	0
$49$	−935.293	−0.0556490
$50$	0	0
$51$	−14114.1	−0.759852
$52$	0	0
$53$	−20343.9	−0.994820	−0.497410	−	0.867516i	$-0.665716\pi$
$53$	−20343.9	−0.994820	−0.497410	+	0.867516i	$0.665716\pi$
$54$	0	0
$55$	−17720.4	−0.789890
$56$	0	0
$57$	4031.64	0.164359
$58$	0	0
$59$	−28488.5	−1.06547	−0.532734	−	0.846283i	$-0.678835\pi$
$59$	−28488.5	−1.06547	−0.532734	+	0.846283i	$0.678835\pi$
$60$	0	0
$61$	16222.3	0.558196	0.279098	−	0.960263i	$-0.409965\pi$
$61$	16222.3	0.558196	0.279098	+	0.960263i	$0.409965\pi$
$62$	0	0
$63$	−10204.6	−0.323926
$64$	0	0
$65$	−91697.4	−2.69199
$66$	0	0
$67$	−54128.0	−1.47311	−0.736555	−	0.676378i	$-0.763549\pi$
$67$	−54128.0	−1.47311	−0.736555	+	0.676378i	$0.763549\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	−42505.6	−1.00069	−0.500346	−	0.865826i	$-0.666794\pi$
$71$	−42505.6	−1.00069	−0.500346	+	0.865826i	$0.666794\pi$
$72$	0	0
$73$	40936.2	0.899084	0.449542	−	0.893259i	$-0.351587\pi$
$73$	40936.2	0.899084	0.449542	+	0.893259i	$0.351587\pi$
$74$	0	0
$75$	−61358.1	−1.25956
$76$	0	0
$77$	22389.1	0.430337
$78$	0	0
$79$	24760.6	0.446369	0.223185	−	0.974776i	$-0.428355\pi$
$79$	24760.6	0.446369	0.223185	+	0.974776i	$0.428355\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	30825.2	0.491147	0.245573	−	0.969378i	$-0.421024\pi$
$83$	30825.2	0.491147	0.245573	+	0.969378i	$0.421024\pi$
$84$	0	0
$85$	156373.	2.34755
$86$	0	0
$87$	11688.4	0.165561
$88$	0	0
$89$	−66531.6	−0.890334	−0.445167	−	0.895448i	$-0.646856\pi$
$89$	−66531.6	−0.890334	−0.445167	+	0.895448i	$0.646856\pi$
$90$	0	0
$91$	115856.	1.46661
$92$	0	0
$93$	−59866.9	−0.717761
$94$	0	0
$95$	−44667.2	−0.507784
$96$	0	0
$97$	−104506.	−1.12774	−0.563872	−	0.825862i	$-0.690689\pi$
$97$	−104506.	−1.12774	−0.563872	+	0.825862i	$0.690689\pi$
$98$	0	0
$99$	−14394.9	−0.147612
$100$	0	0
$101$	−203117.	−1.98127	−0.990635	−	0.136540i	$-0.956402\pi$
$101$	−203117.	−1.98127	−0.990635	+	0.136540i	$0.956402\pi$
$102$	0	0
$103$	148132.	1.37580	0.687901	−	0.725805i	$-0.258532\pi$
$103$	148132.	1.37580	0.687901	+	0.725805i	$0.258532\pi$
$104$	0	0
$105$	113059.	1.00076
$106$	0	0
$107$	43750.9	0.369426	0.184713	−	0.982793i	$-0.440864\pi$
$107$	43750.9	0.369426	0.184713	+	0.982793i	$0.440864\pi$
$108$	0	0
$109$	189052.	1.52410	0.762052	−	0.647516i	$-0.224192\pi$
$109$	189052.	1.52410	0.762052	+	0.647516i	$0.224192\pi$
$110$	0	0
$111$	72511.2	0.558596
$112$	0	0
$113$	−82382.3	−0.606929	−0.303465	−	0.952843i	$-0.598143\pi$
$113$	−82382.3	−0.606929	−0.303465	+	0.952843i	$0.598143\pi$
$114$	0	0
$115$	52747.9	0.371929
$116$	0	0
$117$	−74489.1	−0.503069
$118$	0	0
$119$	−197571.	−1.27896
$120$	0	0
$121$	−129468.	−0.803897
$122$	0	0
$123$	−161473.	−0.962358
$124$	0	0
$125$	368195.	2.10767
$126$	0	0
$127$	95322.4	0.524428	0.262214	−	0.965010i	$-0.415547\pi$
$127$	95322.4	0.524428	0.262214	+	0.965010i	$0.415547\pi$
$128$	0	0
$129$	96396.1	0.510018
$130$	0	0
$131$	−181975.	−0.926474	−0.463237	−	0.886234i	$-0.653312\pi$
$131$	−181975.	−0.926474	−0.463237	+	0.886234i	$0.653312\pi$
$132$	0	0
$133$	56435.3	0.276644
$134$	0	0
$135$	−72690.4	−0.343275
$136$	0	0
$137$	175024.	0.796705	0.398352	−	0.917232i	$-0.369582\pi$
$137$	175024.	0.796705	0.398352	+	0.917232i	$0.369582\pi$
$138$	0	0
$139$	−199672.	−0.876556	−0.438278	−	0.898839i	$-0.644412\pi$
$139$	−199672.	−0.876556	−0.438278	+	0.898839i	$0.644412\pi$
$140$	0	0
$141$	−160982.	−0.681914
$142$	0	0
$143$	163430.	0.668331
$144$	0	0
$145$	−129498.	−0.511497
$146$	0	0
$147$	8417.64	0.0321290
$148$	0	0
$149$	71174.8	0.262640	0.131320	−	0.991340i	$-0.458079\pi$
$149$	71174.8	0.262640	0.131320	+	0.991340i	$0.458079\pi$
$150$	0	0
$151$	−133491.	−0.476440	−0.238220	−	0.971211i	$-0.576564\pi$
$151$	−133491.	−0.476440	−0.238220	+	0.971211i	$0.576564\pi$
$152$	0	0
$153$	127027.	0.438701
$154$	0	0
$155$	663276.	2.21751
$156$	0	0
$157$	−361773.	−1.17135	−0.585675	−	0.810546i	$-0.699171\pi$
$157$	−361773.	−1.17135	−0.585675	+	0.810546i	$0.699171\pi$
$158$	0	0
$159$	183095.	0.574360
$160$	0	0
$161$	−66645.0	−0.202630
$162$	0	0
$163$	627275.	1.84922	0.924611	−	0.380912i	$-0.124390\pi$
$163$	627275.	1.84922	0.924611	+	0.380912i	$0.124390\pi$
$164$	0	0
$165$	159484.	0.456043
$166$	0	0
$167$	−9849.85	−0.0273299	−0.0136650	−	0.999907i	$-0.504350\pi$
$167$	−9849.85	−0.0273299	−0.0136650	+	0.999907i	$0.504350\pi$
$168$	0	0
$169$	474405.	1.27771
$170$	0	0
$171$	−36284.7	−0.0948929
$172$	0	0
$173$	−250617.	−0.636641	−0.318320	−	0.947983i	$-0.603119\pi$
$173$	−250617.	−0.636641	−0.318320	+	0.947983i	$0.603119\pi$
$174$	0	0
$175$	−858898.	−2.12005
$176$	0	0
$177$	256397.	0.615148
$178$	0	0
$179$	−742331.	−1.73167	−0.865835	−	0.500330i	$-0.833212\pi$
$179$	−742331.	−1.73167	−0.865835	+	0.500330i	$0.833212\pi$
$180$	0	0
$181$	−292957.	−0.664673	−0.332336	−	0.943161i	$-0.607837\pi$
$181$	−292957.	−0.664673	−0.332336	+	0.943161i	$0.607837\pi$
$182$	0	0
$183$	−146000.	−0.322275
$184$	0	0
$185$	−803363.	−1.72577
$186$	0	0
$187$	−278699.	−0.582817
$188$	0	0
$189$	91841.6	0.187019
$190$	0	0
$191$	−57674.2	−0.114393	−0.0571963	−	0.998363i	$-0.518216\pi$
$191$	−57674.2	−0.114393	−0.0571963	+	0.998363i	$0.518216\pi$
$192$	0	0
$193$	−178690.	−0.345308	−0.172654	−	0.984983i	$-0.555234\pi$
$193$	−178690.	−0.345308	−0.172654	+	0.984983i	$0.555234\pi$
$194$	0	0
$195$	825276.	1.55422
$196$	0	0
$197$	−525992.	−0.965636	−0.482818	−	0.875721i	$-0.660387\pi$
$197$	−525992.	−0.965636	−0.482818	+	0.875721i	$0.660387\pi$
$198$	0	0
$199$	184283.	0.329877	0.164939	−	0.986304i	$-0.447257\pi$
$199$	184283.	0.329877	0.164939	+	0.986304i	$0.447257\pi$
$200$	0	0
$201$	487152.	0.850500
$202$	0	0
$203$	163616.	0.278667
$204$	0	0
$205$	1.78898e6	2.97318
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	79609.1	0.126066
$210$	0	0
$211$	313460.	0.484703	0.242352	−	0.970188i	$-0.422081\pi$
$211$	313460.	0.484703	0.242352	+	0.970188i	$0.422081\pi$
$212$	0	0
$213$	382550.	0.577749
$214$	0	0
$215$	−1.06799e6	−1.57569
$216$	0	0
$217$	−838024.	−1.20811
$218$	0	0
$219$	−368426.	−0.519086
$220$	0	0
$221$	−1.44218e6	−1.98627
$222$	0	0
$223$	1.00274e6	1.35028	0.675141	−	0.737689i	$-0.264083\pi$
$223$	1.00274e6	1.35028	0.675141	+	0.737689i	$0.264083\pi$
$224$	0	0
$225$	552223.	0.727208
$226$	0	0
$227$	1.02818e6	1.32435	0.662176	−	0.749348i	$-0.269633\pi$
$227$	1.02818e6	1.32435	0.662176	+	0.749348i	$0.269633\pi$
$228$	0	0
$229$	−83843.9	−0.105653	−0.0528266	−	0.998604i	$-0.516823\pi$
$229$	−83843.9	−0.105653	−0.0528266	+	0.998604i	$0.516823\pi$
$230$	0	0
$231$	−201502.	−0.248455
$232$	0	0
$233$	−241626.	−0.291577	−0.145789	−	0.989316i	$-0.546572\pi$
$233$	−241626.	−0.291577	−0.145789	+	0.989316i	$0.546572\pi$
$234$	0	0
$235$	1.78355e6	2.10676
$236$	0	0
$237$	−222846.	−0.257711
$238$	0	0
$239$	−1.32779e6	−1.50361	−0.751806	−	0.659384i	$-0.770817\pi$
$239$	−1.32779e6	−1.50361	−0.751806	+	0.659384i	$0.770817\pi$
$240$	0	0
$241$	−1.21493e6	−1.34744	−0.673721	−	0.738986i	$-0.735305\pi$
$241$	−1.21493e6	−1.34744	−0.673721	+	0.738986i	$0.735305\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−93260.3	−0.0992617
$246$	0	0
$247$	411952.	0.429639
$248$	0	0
$249$	−277427.	−0.283564
$250$	0	0
$251$	−1.21766e6	−1.21995	−0.609977	−	0.792419i	$-0.708821\pi$
$251$	−1.21766e6	−1.21995	−0.609977	+	0.792419i	$0.708821\pi$
$252$	0	0
$253$	−94011.2	−0.0923376
$254$	0	0
$255$	−1.40736e6	−1.35536
$256$	0	0
$257$	1.21439e6	1.14690	0.573450	−	0.819241i	$-0.305605\pi$
$257$	1.21439e6	1.14690	0.573450	+	0.819241i	$0.305605\pi$
$258$	0	0
$259$	1.01502e6	0.940210
$260$	0	0
$261$	−105196.	−0.0955867
$262$	0	0
$263$	−93687.8	−0.0835206	−0.0417603	−	0.999128i	$-0.513297\pi$
$263$	−93687.8	−0.0835206	−0.0417603	+	0.999128i	$0.513297\pi$
$264$	0	0
$265$	−2.02854e6	−1.77447
$266$	0	0
$267$	598784.	0.514034
$268$	0	0
$269$	−323647.	−0.272703	−0.136352	−	0.990660i	$-0.543538\pi$
$269$	−323647.	−0.272703	−0.136352	+	0.990660i	$0.543538\pi$
$270$	0	0
$271$	−1.76235e6	−1.45771	−0.728853	−	0.684670i	$-0.759946\pi$
$271$	−1.76235e6	−1.45771	−0.728853	+	0.684670i	$0.759946\pi$
$272$	0	0
$273$	−1.04271e6	−0.846750
$274$	0	0
$275$	−1.21158e6	−0.966100
$276$	0	0
$277$	−1.83478e6	−1.43676	−0.718381	−	0.695650i	$-0.755117\pi$
$277$	−1.83478e6	−1.43676	−0.718381	+	0.695650i	$0.755117\pi$
$278$	0	0
$279$	538803.	0.414399
$280$	0	0
$281$	−1.71252e6	−1.29381	−0.646903	−	0.762572i	$-0.723936\pi$
$281$	−1.71252e6	−1.29381	−0.646903	+	0.762572i	$0.723936\pi$
$282$	0	0
$283$	−1.53094e6	−1.13629	−0.568147	−	0.822927i	$-0.692340\pi$
$283$	−1.53094e6	−1.13629	−0.568147	+	0.822927i	$0.692340\pi$
$284$	0	0
$285$	402004.	0.293169
$286$	0	0
$287$	−2.26031e6	−1.61981
$288$	0	0
$289$	1.03951e6	0.732125
$290$	0	0
$291$	940550.	0.651103
$292$	0	0
$293$	−2.02997e6	−1.38140	−0.690701	−	0.723141i	$-0.742698\pi$
$293$	−2.02997e6	−1.38140	−0.690701	+	0.723141i	$0.742698\pi$
$294$	0	0
$295$	−2.84066e6	−1.90049
$296$	0	0
$297$	129554.	0.0852237
$298$	0	0
$299$	−486478.	−0.314692
$300$	0	0
$301$	1.34936e6	0.858445
$302$	0	0
$303$	1.82806e6	1.14389
$304$	0	0
$305$	1.61756e6	0.995661
$306$	0	0
$307$	−396907.	−0.240349	−0.120175	−	0.992753i	$-0.538345\pi$
$307$	−396907.	−0.240349	−0.120175	+	0.992753i	$0.538345\pi$
$308$	0	0
$309$	−1.33319e6	−0.794319
$310$	0	0
$311$	−1.27993e6	−0.750385	−0.375192	−	0.926947i	$-0.622423\pi$
$311$	−1.27993e6	−0.750385	−0.375192	+	0.926947i	$0.622423\pi$
$312$	0	0
$313$	−2.41300e6	−1.39218	−0.696091	−	0.717953i	$-0.745079\pi$
$313$	−2.41300e6	−1.39218	−0.696091	+	0.717953i	$0.745079\pi$
$314$	0	0
$315$	−1.01753e6	−0.577790
$316$	0	0
$317$	1.03063e6	0.576043	0.288022	−	0.957624i	$-0.407002\pi$
$317$	1.03063e6	0.576043	0.288022	+	0.957624i	$0.407002\pi$
$318$	0	0
$319$	230801.	0.126988
$320$	0	0
$321$	−393758.	−0.213288
$322$	0	0
$323$	−702507.	−0.374666
$324$	0	0
$325$	−6.26956e6	−3.29252
$326$	0	0
$327$	−1.70147e6	−0.879942
$328$	0	0
$329$	−2.25345e6	−1.14778
$330$	0	0
$331$	−231656.	−0.116218	−0.0581089	−	0.998310i	$-0.518507\pi$
$331$	−231656.	−0.116218	−0.0581089	+	0.998310i	$0.518507\pi$
$332$	0	0
$333$	−652601.	−0.322505
$334$	0	0
$335$	−5.39723e6	−2.62760
$336$	0	0
$337$	−351073.	−0.168393	−0.0841963	−	0.996449i	$-0.526832\pi$
$337$	−351073.	−0.168393	−0.0841963	+	0.996449i	$0.526832\pi$
$338$	0	0
$339$	741441.	0.350411
$340$	0	0
$341$	−1.18214e6	−0.550532
$342$	0	0
$343$	2.23523e6	1.02586
$344$	0	0
$345$	−474731.	−0.214734
$346$	0	0
$347$	550941.	0.245630	0.122815	−	0.992430i	$-0.460808\pi$
$347$	550941.	0.245630	0.122815	+	0.992430i	$0.460808\pi$
$348$	0	0
$349$	−1.30593e6	−0.573928	−0.286964	−	0.957941i	$-0.592646\pi$
$349$	−1.30593e6	−0.573928	−0.286964	+	0.957941i	$0.592646\pi$
$350$	0	0
$351$	670402.	0.290447
$352$	0	0
$353$	2.13442e6	0.911682	0.455841	−	0.890061i	$-0.349339\pi$
$353$	2.13442e6	0.911682	0.455841	+	0.890061i	$0.349339\pi$
$354$	0	0
$355$	−4.23834e6	−1.78494
$356$	0	0
$357$	1.77814e6	0.738407
$358$	0	0
$359$	−1.06632e6	−0.436670	−0.218335	−	0.975874i	$-0.570063\pi$
$359$	−1.06632e6	−0.436670	−0.218335	+	0.975874i	$0.570063\pi$
$360$	0	0
$361$	−2.27543e6	−0.918958
$362$	0	0
$363$	1.16522e6	0.464130
$364$	0	0
$365$	4.08185e6	1.60371
$366$	0	0
$367$	−1.34294e6	−0.520464	−0.260232	−	0.965546i	$-0.583799\pi$
$367$	−1.34294e6	−0.520464	−0.260232	+	0.965546i	$0.583799\pi$
$368$	0	0
$369$	1.45326e6	0.555618
$370$	0	0
$371$	2.56299e6	0.966744
$372$	0	0
$373$	3.22658e6	1.20080	0.600400	−	0.799700i	$-0.295008\pi$
$373$	3.22658e6	1.20080	0.600400	+	0.799700i	$0.295008\pi$
$374$	0	0
$375$	−3.31376e6	−1.21687
$376$	0	0
$377$	1.19432e6	0.432781
$378$	0	0
$379$	2.41939e6	0.865184	0.432592	−	0.901590i	$-0.357599\pi$
$379$	2.41939e6	0.865184	0.432592	+	0.901590i	$0.357599\pi$
$380$	0	0
$381$	−857902.	−0.302779
$382$	0	0
$383$	−1.03280e6	−0.359764	−0.179882	−	0.983688i	$-0.557572\pi$
$383$	−1.03280e6	−0.359764	−0.179882	+	0.983688i	$0.557572\pi$
$384$	0	0
$385$	2.23247e6	0.767597
$386$	0	0
$387$	−867565.	−0.294459
$388$	0	0
$389$	2.56135e6	0.858212	0.429106	−	0.903254i	$-0.358829\pi$
$389$	2.56135e6	0.858212	0.429106	+	0.903254i	$0.358829\pi$
$390$	0	0
$391$	829598.	0.274426
$392$	0	0
$393$	1.63777e6	0.534900
$394$	0	0
$395$	2.46894e6	0.796193
$396$	0	0
$397$	−2.07970e6	−0.662255	−0.331128	−	0.943586i	$-0.607429\pi$
$397$	−2.07970e6	−0.662255	−0.331128	+	0.943586i	$0.607429\pi$
$398$	0	0
$399$	−507918.	−0.159721
$400$	0	0
$401$	−1.97329e6	−0.612817	−0.306408	−	0.951900i	$-0.599127\pi$
$401$	−1.97329e6	−0.612817	−0.306408	+	0.951900i	$0.599127\pi$
$402$	0	0
$403$	−6.11719e6	−1.87624
$404$	0	0
$405$	654213.	0.198190
$406$	0	0
$407$	1.43181e6	0.428450
$408$	0	0
$409$	1.68806e6	0.498975	0.249488	−	0.968378i	$-0.419738\pi$
$409$	1.68806e6	0.498975	0.249488	+	0.968378i	$0.419738\pi$
$410$	0	0
$411$	−1.57522e6	−0.459978
$412$	0	0
$413$	3.58907e6	1.03540
$414$	0	0
$415$	3.07366e6	0.876064
$416$	0	0
$417$	1.79705e6	0.506080
$418$	0	0
$419$	−1.72868e6	−0.481039	−0.240520	−	0.970644i	$-0.577318\pi$
$419$	−1.72868e6	−0.481039	−0.240520	+	0.970644i	$0.577318\pi$
$420$	0	0
$421$	932870.	0.256517	0.128258	−	0.991741i	$-0.459061\pi$
$421$	932870.	0.256517	0.128258	+	0.991741i	$0.459061\pi$
$422$	0	0
$423$	1.44884e6	0.393703
$424$	0	0
$425$	1.06916e7	2.87124
$426$	0	0
$427$	−2.04373e6	−0.542443
$428$	0	0
$429$	−1.47087e6	−0.385861
$430$	0	0
$431$	7.62939e6	1.97832	0.989161	−	0.146838i	$-0.0469095\pi$
$431$	7.62939e6	1.97832	0.989161	+	0.146838i	$0.0469095\pi$
$432$	0	0
$433$	4.68013e6	1.19960	0.599802	−	0.800148i	$-0.295246\pi$
$433$	4.68013e6	1.19960	0.599802	+	0.800148i	$0.295246\pi$
$434$	0	0
$435$	1.16548e6	0.295313
$436$	0	0
$437$	−236971.	−0.0593596
$438$	0	0
$439$	−1.21981e6	−0.302087	−0.151043	−	0.988527i	$-0.548263\pi$
$439$	−1.21981e6	−0.302087	−0.151043	+	0.988527i	$0.548263\pi$
$440$	0	0
$441$	−75758.7	−0.0185497
$442$	0	0
$443$	1.82127e6	0.440926	0.220463	−	0.975395i	$-0.429243\pi$
$443$	1.82127e6	0.440926	0.220463	+	0.975395i	$0.429243\pi$
$444$	0	0
$445$	−6.63403e6	−1.58810
$446$	0	0
$447$	−640573.	−0.151635
$448$	0	0
$449$	−3.31712e6	−0.776506	−0.388253	−	0.921553i	$-0.626921\pi$
$449$	−3.31712e6	−0.776506	−0.388253	+	0.921553i	$0.626921\pi$
$450$	0	0
$451$	−3.18846e6	−0.738142
$452$	0	0
$453$	1.20142e6	0.275073
$454$	0	0
$455$	1.15523e7	2.61602
$456$	0	0
$457$	−2.14897e6	−0.481326	−0.240663	−	0.970609i	$-0.577365\pi$
$457$	−2.14897e6	−0.481326	−0.240663	+	0.970609i	$0.577365\pi$
$458$	0	0
$459$	−1.14325e6	−0.253284
$460$	0	0
$461$	−6.83686e6	−1.49832	−0.749160	−	0.662389i	$-0.769543\pi$
$461$	−6.83686e6	−1.49832	−0.749160	+	0.662389i	$0.769543\pi$
$462$	0	0
$463$	−3.26293e6	−0.707385	−0.353692	−	0.935362i	$-0.615074\pi$
$463$	−3.26293e6	−0.707385	−0.353692	+	0.935362i	$0.615074\pi$
$464$	0	0
$465$	−5.96948e6	−1.28028
$466$	0	0
$467$	4.30238e6	0.912886	0.456443	−	0.889753i	$-0.349123\pi$
$467$	4.30238e6	0.912886	0.456443	+	0.889753i	$0.349123\pi$
$468$	0	0
$469$	6.81920e6	1.43153
$470$	0	0
$471$	3.25595e6	0.676279
$472$	0	0
$473$	1.90345e6	0.391191
$474$	0	0
$475$	−3.05400e6	−0.621061
$476$	0	0
$477$	−1.64786e6	−0.331607
$478$	0	0
$479$	−3.15612e6	−0.628514	−0.314257	−	0.949338i	$-0.601755\pi$
$479$	−3.15612e6	−0.628514	−0.314257	+	0.949338i	$0.601755\pi$
$480$	0	0
$481$	7.40918e6	1.46018
$482$	0	0
$483$	599805.	0.116988
$484$	0	0
$485$	−1.04205e7	−2.01157
$486$	0	0
$487$	1.04981e6	0.200581	0.100290	−	0.994958i	$-0.468023\pi$
$487$	1.04981e6	0.200581	0.100290	+	0.994958i	$0.468023\pi$
$488$	0	0
$489$	−5.64548e6	−1.06765
$490$	0	0
$491$	−1.67278e6	−0.313138	−0.156569	−	0.987667i	$-0.550043\pi$
$491$	−1.67278e6	−0.313138	−0.156569	+	0.987667i	$0.550043\pi$
$492$	0	0
$493$	−2.03669e6	−0.377406
$494$	0	0
$495$	−1.43535e6	−0.263297
$496$	0	0
$497$	5.35498e6	0.972449
$498$	0	0
$499$	3.07979e6	0.553695	0.276847	−	0.960914i	$-0.410710\pi$
$499$	3.07979e6	0.553695	0.276847	+	0.960914i	$0.410710\pi$
$500$	0	0
$501$	88648.7	0.0157789
$502$	0	0
$503$	−4.06934e6	−0.717140	−0.358570	−	0.933503i	$-0.616736\pi$
$503$	−4.06934e6	−0.717140	−0.358570	+	0.933503i	$0.616736\pi$
$504$	0	0
$505$	−2.02533e7	−3.53401
$506$	0	0
$507$	−4.26964e6	−0.737686
$508$	0	0
$509$	5.65269e6	0.967077	0.483538	−	0.875323i	$-0.339351\pi$
$509$	5.65269e6	0.967077	0.483538	+	0.875323i	$0.339351\pi$
$510$	0	0
$511$	−5.15726e6	−0.873709
$512$	0	0
$513$	326563.	0.0547864
$514$	0	0
$515$	1.47706e7	2.45403
$516$	0	0
$517$	−3.17877e6	−0.523038
$518$	0	0
$519$	2.25555e6	0.367565
$520$	0	0
$521$	−729056.	−0.117670	−0.0588351	−	0.998268i	$-0.518739\pi$
$521$	−729056.	−0.117670	−0.0588351	+	0.998268i	$0.518739\pi$
$522$	0	0
$523$	−5.74086e6	−0.917747	−0.458873	−	0.888502i	$-0.651747\pi$
$523$	−5.74086e6	−0.917747	−0.458873	+	0.888502i	$0.651747\pi$
$524$	0	0
$525$	7.73008e6	1.22401
$526$	0	0
$527$	1.04317e7	1.63618
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	−2.30757e6	−0.355156
$532$	0	0
$533$	−1.64993e7	−2.51563
$534$	0	0
$535$	4.36250e6	0.658948
$536$	0	0
$537$	6.68098e6	0.999780
$538$	0	0
$539$	166216.	0.0246434
$540$	0	0
$541$	1.12666e7	1.65500	0.827500	−	0.561466i	$-0.189762\pi$
$541$	1.12666e7	1.65500	0.827500	+	0.561466i	$0.189762\pi$
$542$	0	0
$543$	2.63661e6	0.383749
$544$	0	0
$545$	1.88508e7	2.71856
$546$	0	0
$547$	−1.92802e6	−0.275513	−0.137757	−	0.990466i	$-0.543989\pi$
$547$	−1.92802e6	−0.275513	−0.137757	+	0.990466i	$0.543989\pi$
$548$	0	0
$549$	1.31400e6	0.186065
$550$	0	0
$551$	581772.	0.0816345
$552$	0	0
$553$	−3.11942e6	−0.433771
$554$	0	0
$555$	7.23027e6	0.996373
$556$	0	0
$557$	−1.27385e7	−1.73973	−0.869865	−	0.493290i	$-0.835794\pi$
$557$	−1.27385e7	−1.73973	−0.869865	+	0.493290i	$0.835794\pi$
$558$	0	0
$559$	9.84974e6	1.33320
$560$	0	0
$561$	2.50829e6	0.336489
$562$	0	0
$563$	−1.30970e7	−1.74140	−0.870702	−	0.491810i	$-0.836335\pi$
$563$	−1.30970e7	−1.74140	−0.870702	+	0.491810i	$0.836335\pi$
$564$	0	0
$565$	−8.21454e6	−1.08259
$566$	0	0
$567$	−826574.	−0.107975
$568$	0	0
$569$	1.32454e7	1.71509	0.857543	−	0.514413i	$-0.171990\pi$
$569$	1.32454e7	1.71509	0.857543	+	0.514413i	$0.171990\pi$
$570$	0	0
$571$	4.22643e6	0.542480	0.271240	−	0.962512i	$-0.412566\pi$
$571$	4.22643e6	0.542480	0.271240	+	0.962512i	$0.412566\pi$
$572$	0	0
$573$	519068.	0.0660446
$574$	0	0
$575$	3.60650e6	0.454900
$576$	0	0
$577$	−224943.	−0.0281276	−0.0140638	−	0.999901i	$-0.504477\pi$
$577$	−224943.	−0.0281276	−0.0140638	+	0.999901i	$0.504477\pi$
$578$	0	0
$579$	1.60821e6	0.199364
$580$	0	0
$581$	−3.88346e6	−0.477285
$582$	0	0
$583$	3.61542e6	0.440542
$584$	0	0
$585$	−7.42749e6	−0.897330
$586$	0	0
$587$	−4.80198e6	−0.575209	−0.287604	−	0.957749i	$-0.592859\pi$
$587$	−4.80198e6	−0.575209	−0.287604	+	0.957749i	$0.592859\pi$
$588$	0	0
$589$	−2.97978e6	−0.353912
$590$	0	0
$591$	4.73392e6	0.557510
$592$	0	0
$593$	−9.07891e6	−1.06022	−0.530111	−	0.847928i	$-0.677850\pi$
$593$	−9.07891e6	−1.06022	−0.530111	+	0.847928i	$0.677850\pi$
$594$	0	0
$595$	−1.97003e7	−2.28129
$596$	0	0
$597$	−1.65855e6	−0.190455
$598$	0	0
$599$	487665.	0.0555334	0.0277667	−	0.999614i	$-0.491160\pi$
$599$	487665.	0.0555334	0.0277667	+	0.999614i	$0.491160\pi$
$600$	0	0
$601$	−9.72337e6	−1.09807	−0.549036	−	0.835799i	$-0.685005\pi$
$601$	−9.72337e6	−1.09807	−0.549036	+	0.835799i	$0.685005\pi$
$602$	0	0
$603$	−4.38437e6	−0.491036
$604$	0	0
$605$	−1.29096e7	−1.43392
$606$	0	0
$607$	3.30773e6	0.364384	0.182192	−	0.983263i	$-0.441681\pi$
$607$	3.30773e6	0.364384	0.182192	+	0.983263i	$0.441681\pi$
$608$	0	0
$609$	−1.47254e6	−0.160888
$610$	0	0
$611$	−1.64491e7	−1.78254
$612$	0	0
$613$	−234065.	−0.0251585	−0.0125793	−	0.999921i	$-0.504004\pi$
$613$	−234065.	−0.0251585	−0.0125793	+	0.999921i	$0.504004\pi$
$614$	0	0
$615$	−1.61009e7	−1.71657
$616$	0	0
$617$	−9.19095e6	−0.971958	−0.485979	−	0.873970i	$-0.661537\pi$
$617$	−9.19095e6	−0.971958	−0.485979	+	0.873970i	$0.661537\pi$
$618$	0	0
$619$	−7.15853e6	−0.750926	−0.375463	−	0.926837i	$-0.622516\pi$
$619$	−7.15853e6	−0.750926	−0.375463	+	0.926837i	$0.622516\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	8.38185e6	0.865206
$624$	0	0
$625$	1.54088e7	1.57786
$626$	0	0
$627$	−716482.	−0.0727841
$628$	0	0
$629$	−1.26350e7	−1.27335
$630$	0	0
$631$	−1.00474e7	−1.00457	−0.502283	−	0.864703i	$-0.667507\pi$
$631$	−1.00474e7	−1.00457	−0.502283	+	0.864703i	$0.667507\pi$
$632$	0	0
$633$	−2.82114e6	−0.279844
$634$	0	0
$635$	9.50483e6	0.935427
$636$	0	0
$637$	860112.	0.0839859
$638$	0	0
$639$	−3.44295e6	−0.333564
$640$	0	0
$641$	4.21517e6	0.405201	0.202600	−	0.979262i	$-0.435061\pi$
$641$	4.21517e6	0.405201	0.202600	+	0.979262i	$0.435061\pi$
$642$	0	0
$643$	−9.48561e6	−0.904770	−0.452385	−	0.891823i	$-0.649427\pi$
$643$	−9.48561e6	−0.904770	−0.452385	+	0.891823i	$0.649427\pi$
$644$	0	0
$645$	9.61189e6	0.909724
$646$	0	0
$647$	−2.54033e6	−0.238578	−0.119289	−	0.992860i	$-0.538061\pi$
$647$	−2.54033e6	−0.238578	−0.119289	+	0.992860i	$0.538061\pi$
$648$	0	0
$649$	5.06284e6	0.471827
$650$	0	0
$651$	7.54222e6	0.697504
$652$	0	0
$653$	9.06671e6	0.832083	0.416042	−	0.909346i	$-0.363417\pi$
$653$	9.06671e6	0.832083	0.416042	+	0.909346i	$0.363417\pi$
$654$	0	0
$655$	−1.81452e7	−1.65256
$656$	0	0
$657$	3.31583e6	0.299695
$658$	0	0
$659$	1.63938e7	1.47050	0.735251	−	0.677795i	$-0.237064\pi$
$659$	1.63938e7	1.47050	0.735251	+	0.677795i	$0.237064\pi$
$660$	0	0
$661$	−1.06203e7	−0.945435	−0.472717	−	0.881214i	$-0.656727\pi$
$661$	−1.06203e7	−0.945435	−0.472717	+	0.881214i	$0.656727\pi$
$662$	0	0
$663$	1.29796e7	1.14677
$664$	0	0
$665$	5.62730e6	0.493453
$666$	0	0
$667$	−687020.	−0.0597936
$668$	0	0
$669$	−9.02462e6	−0.779586
$670$	0	0
$671$	−2.88294e6	−0.247189
$672$	0	0
$673$	−1.19056e7	−1.01324	−0.506620	−	0.862169i	$-0.669105\pi$
$673$	−1.19056e7	−1.01324	−0.506620	+	0.862169i	$0.669105\pi$
$674$	0	0
$675$	−4.97001e6	−0.419854
$676$	0	0
$677$	2.17878e7	1.82701	0.913507	−	0.406822i	$-0.133363\pi$
$677$	2.17878e7	1.82701	0.913507	+	0.406822i	$0.133363\pi$
$678$	0	0
$679$	1.31659e7	1.09592
$680$	0	0
$681$	−9.25360e6	−0.764615
$682$	0	0
$683$	−8.27503e6	−0.678762	−0.339381	−	0.940649i	$-0.610218\pi$
$683$	−8.27503e6	−0.678762	−0.339381	+	0.940649i	$0.610218\pi$
$684$	0	0
$685$	1.74521e7	1.42109
$686$	0	0
$687$	754595.	0.0609989
$688$	0	0
$689$	1.87086e7	1.50139
$690$	0	0
$691$	−8.37140e6	−0.666965	−0.333483	−	0.942756i	$-0.608224\pi$
$691$	−8.37140e6	−0.666965	−0.333483	+	0.942756i	$0.608224\pi$
$692$	0	0
$693$	1.81351e6	0.143446
$694$	0	0
$695$	−1.99098e7	−1.56352
$696$	0	0
$697$	2.81364e7	2.19375
$698$	0	0
$699$	2.17463e6	0.168342
$700$	0	0
$701$	3.01421e6	0.231675	0.115837	−	0.993268i	$-0.463045\pi$
$701$	3.01421e6	0.231675	0.115837	+	0.993268i	$0.463045\pi$
$702$	0	0
$703$	3.60912e6	0.275431
$704$	0	0
$705$	−1.60519e7	−1.21634
$706$	0	0
$707$	2.55893e7	1.92535
$708$	0	0
$709$	−5.74628e6	−0.429310	−0.214655	−	0.976690i	$-0.568863\pi$
$709$	−5.74628e6	−0.429310	−0.214655	+	0.976690i	$0.568863\pi$
$710$	0	0
$711$	2.00561e6	0.148790
$712$	0	0
$713$	3.51885e6	0.259225
$714$	0	0
$715$	1.62960e7	1.19211
$716$	0	0
$717$	1.19501e7	0.868111
$718$	0	0
$719$	1.64352e7	1.18564	0.592821	−	0.805334i	$-0.298014\pi$
$719$	1.64352e7	1.18564	0.592821	+	0.805334i	$0.298014\pi$
$720$	0	0
$721$	−1.86621e7	−1.33697
$722$	0	0
$723$	1.09344e7	0.777945
$724$	0	0
$725$	−8.85408e6	−0.625603
$726$	0	0
$727$	4.65331e6	0.326532	0.163266	−	0.986582i	$-0.447797\pi$
$727$	4.65331e6	0.326532	0.163266	+	0.986582i	$0.447797\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.67969e7	−1.16261
$732$	0	0
$733$	1.76748e7	1.21505	0.607527	−	0.794299i	$-0.292162\pi$
$733$	1.76748e7	1.21505	0.607527	+	0.794299i	$0.292162\pi$
$734$	0	0
$735$	839343.	0.0573088
$736$	0	0
$737$	9.61936e6	0.652345
$738$	0	0
$739$	−2.32944e7	−1.56907	−0.784533	−	0.620087i	$-0.787097\pi$
$739$	−2.32944e7	−1.56907	−0.784533	+	0.620087i	$0.787097\pi$
$740$	0	0
$741$	−3.70757e6	−0.248052
$742$	0	0
$743$	−1.77107e7	−1.17696	−0.588482	−	0.808511i	$-0.700274\pi$
$743$	−1.77107e7	−1.17696	−0.588482	+	0.808511i	$0.700274\pi$
$744$	0	0
$745$	7.09701e6	0.468473
$746$	0	0
$747$	2.49685e6	0.163716
$748$	0	0
$749$	−5.51186e6	−0.358999
$750$	0	0
$751$	9.59704e6	0.620922	0.310461	−	0.950586i	$-0.399517\pi$
$751$	9.59704e6	0.620922	0.310461	+	0.950586i	$0.399517\pi$
$752$	0	0
$753$	1.09590e7	0.704340
$754$	0	0
$755$	−1.33107e7	−0.849832
$756$	0	0
$757$	1.81520e7	1.15129	0.575645	−	0.817700i	$-0.304751\pi$
$757$	1.81520e7	1.15129	0.575645	+	0.817700i	$0.304751\pi$
$758$	0	0
$759$	846101.	0.0533111
$760$	0	0
$761$	2.18570e7	1.36813	0.684067	−	0.729419i	$-0.260210\pi$
$761$	2.18570e7	1.36813	0.684067	+	0.729419i	$0.260210\pi$
$762$	0	0
$763$	−2.38173e7	−1.48109
$764$	0	0
$765$	1.26662e7	0.782515
$766$	0	0
$767$	2.61986e7	1.60801
$768$	0	0
$769$	−3.01818e6	−0.184047	−0.0920237	−	0.995757i	$-0.529334\pi$
$769$	−3.01818e6	−0.184047	−0.0920237	+	0.995757i	$0.529334\pi$
$770$	0	0
$771$	−1.09295e7	−0.662163
$772$	0	0
$773$	7.09462e6	0.427052	0.213526	−	0.976937i	$-0.431505\pi$
$773$	7.09462e6	0.427052	0.213526	+	0.976937i	$0.431505\pi$
$774$	0	0
$775$	4.53497e7	2.71219
$776$	0	0
$777$	−9.13517e6	−0.542830
$778$	0	0
$779$	−8.03704e6	−0.474518
$780$	0	0
$781$	7.55388e6	0.443142
$782$	0	0
$783$	946763.	0.0551870
$784$	0	0
$785$	−3.60732e7	−2.08935
$786$	0	0
$787$	−2.85125e7	−1.64096	−0.820482	−	0.571672i	$-0.806295\pi$
$787$	−2.85125e7	−1.64096	−0.820482	+	0.571672i	$0.806295\pi$
$788$	0	0
$789$	843190.	0.0482206
$790$	0	0
$791$	1.03788e7	0.589800
$792$	0	0
$793$	−1.49183e7	−0.842435
$794$	0	0
$795$	1.82569e7	1.02449
$796$	0	0
$797$	7.59475e6	0.423514	0.211757	−	0.977322i	$-0.432081\pi$
$797$	7.59475e6	0.423514	0.211757	+	0.977322i	$0.432081\pi$
$798$	0	0
$799$	2.80509e7	1.55446
$800$	0	0
$801$	−5.38906e6	−0.296778
$802$	0	0
$803$	−7.27497e6	−0.398146
$804$	0	0
$805$	−6.64533e6	−0.361432
$806$	0	0
$807$	2.91282e6	0.157445
$808$	0	0
$809$	135394.	0.00727324	0.00363662	−	0.999993i	$-0.498842\pi$
$809$	135394.	0.00727324	0.00363662	+	0.999993i	$0.498842\pi$
$810$	0	0
$811$	2.84394e7	1.51834	0.759168	−	0.650895i	$-0.225606\pi$
$811$	2.84394e7	1.51834	0.759168	+	0.650895i	$0.225606\pi$
$812$	0	0
$813$	1.58612e7	0.841607
$814$	0	0
$815$	6.25471e7	3.29848
$816$	0	0
$817$	4.79795e6	0.251479
$818$	0	0
$819$	9.38435e6	0.488871
$820$	0	0
$821$	−9.25492e6	−0.479198	−0.239599	−	0.970872i	$-0.577016\pi$
$821$	−9.25492e6	−0.479198	−0.239599	+	0.970872i	$0.577016\pi$
$822$	0	0
$823$	−8.81289e6	−0.453543	−0.226772	−	0.973948i	$-0.572817\pi$
$823$	−8.81289e6	−0.453543	−0.226772	+	0.973948i	$0.572817\pi$
$824$	0	0
$825$	1.09043e7	0.557778
$826$	0	0
$827$	−2.40557e7	−1.22308	−0.611539	−	0.791214i	$-0.709449\pi$
$827$	−2.40557e7	−1.22308	−0.611539	+	0.791214i	$0.709449\pi$
$828$	0	0
$829$	3.34899e7	1.69249	0.846247	−	0.532791i	$-0.178857\pi$
$829$	3.34899e7	1.69249	0.846247	+	0.532791i	$0.178857\pi$
$830$	0	0
$831$	1.65130e7	0.829515
$832$	0	0
$833$	−1.46676e6	−0.0732398
$834$	0	0
$835$	−982153.	−0.0487487
$836$	0	0
$837$	−4.84922e6	−0.239254
$838$	0	0
$839$	−6.66922e6	−0.327092	−0.163546	−	0.986536i	$-0.552293\pi$
$839$	−6.66922e6	−0.327092	−0.163546	+	0.986536i	$0.552293\pi$
$840$	0	0
$841$	−1.88245e7	−0.917769
$842$	0	0
$843$	1.54127e7	0.746979
$844$	0	0
$845$	4.73040e7	2.27906
$846$	0	0
$847$	1.63108e7	0.781209
$848$	0	0
$849$	1.37784e7	0.656040
$850$	0	0
$851$	−4.26205e6	−0.201741
$852$	0	0
$853$	2.91274e6	0.137066	0.0685330	−	0.997649i	$-0.478168\pi$
$853$	2.91274e6	0.137066	0.0685330	+	0.997649i	$0.478168\pi$
$854$	0	0
$855$	−3.61804e6	−0.169261
$856$	0	0
$857$	1.52102e7	0.707429	0.353715	−	0.935353i	$-0.384918\pi$
$857$	1.52102e7	0.707429	0.353715	+	0.935353i	$0.384918\pi$
$858$	0	0
$859$	−1.06621e7	−0.493016	−0.246508	−	0.969141i	$-0.579283\pi$
$859$	−1.06621e7	−0.493016	−0.246508	+	0.969141i	$0.579283\pi$
$860$	0	0
$861$	2.03428e7	0.935198
$862$	0	0
$863$	3.79585e7	1.73493	0.867465	−	0.497498i	$-0.165748\pi$
$863$	3.79585e7	1.73493	0.867465	+	0.497498i	$0.165748\pi$
$864$	0	0
$865$	−2.49896e7	−1.13558
$866$	0	0
$867$	−9.35562e6	−0.422693
$868$	0	0
$869$	−4.40034e6	−0.197668
$870$	0	0
$871$	4.97771e7	2.22323
$872$	0	0
$873$	−8.46495e6	−0.375914
$874$	0	0
$875$	−4.63863e7	−2.04819
$876$	0	0
$877$	2.72329e7	1.19562	0.597812	−	0.801636i	$-0.296037\pi$
$877$	2.72329e7	1.19562	0.597812	+	0.801636i	$0.296037\pi$
$878$	0	0
$879$	1.82697e7	0.797553
$880$	0	0
$881$	8.49018e6	0.368534	0.184267	−	0.982876i	$-0.441009\pi$
$881$	8.49018e6	0.368534	0.184267	+	0.982876i	$0.441009\pi$
$882$	0	0
$883$	2.58940e7	1.11763	0.558815	−	0.829292i	$-0.311256\pi$
$883$	2.58940e7	1.11763	0.558815	+	0.829292i	$0.311256\pi$
$884$	0	0
$885$	2.55660e7	1.09725
$886$	0	0
$887$	1.57892e7	0.673832	0.336916	−	0.941535i	$-0.390616\pi$
$887$	1.57892e7	0.673832	0.336916	+	0.941535i	$0.390616\pi$
$888$	0	0
$889$	−1.20090e7	−0.509627
$890$	0	0
$891$	−1.16599e6	−0.0492039
$892$	0	0
$893$	−8.01261e6	−0.336237
$894$	0	0
$895$	−7.40196e7	−3.08880
$896$	0	0
$897$	4.37830e6	0.181687
$898$	0	0
$899$	−8.63890e6	−0.356500
$900$	0	0
$901$	−3.19041e7	−1.30929
$902$	0	0
$903$	−1.21443e7	−0.495624
$904$	0	0
$905$	−2.92115e7	−1.18558
$906$	0	0
$907$	1.12308e7	0.453305	0.226653	−	0.973976i	$-0.427222\pi$
$907$	1.12308e7	0.453305	0.226653	+	0.973976i	$0.427222\pi$
$908$	0	0
$909$	−1.64525e7	−0.660423
$910$	0	0
$911$	−2.61794e7	−1.04511	−0.522557	−	0.852604i	$-0.675022\pi$
$911$	−2.61794e7	−1.04511	−0.522557	+	0.852604i	$0.675022\pi$
$912$	0	0
$913$	−5.47811e6	−0.217497
$914$	0	0
$915$	−1.45581e7	−0.574845
$916$	0	0
$917$	2.29257e7	0.900326
$918$	0	0
$919$	−4.07119e7	−1.59013	−0.795065	−	0.606524i	$-0.792563\pi$
$919$	−4.07119e7	−1.59013	−0.795065	+	0.606524i	$0.792563\pi$
$920$	0	0
$921$	3.57217e6	0.138766
$922$	0	0
$923$	3.90889e7	1.51025
$924$	0	0
$925$	−5.49278e7	−2.11075
$926$	0	0
$927$	1.19987e7	0.458600
$928$	0	0
$929$	8.89152e6	0.338015	0.169008	−	0.985615i	$-0.445944\pi$
$929$	8.89152e6	0.338015	0.169008	+	0.985615i	$0.445944\pi$
$930$	0	0
$931$	418973.	0.0158421
$932$	0	0
$933$	1.15193e7	0.433235
$934$	0	0
$935$	−2.77898e7	−1.03958
$936$	0	0
$937$	4.07318e6	0.151560	0.0757799	−	0.997125i	$-0.475855\pi$
$937$	4.07318e6	0.151560	0.0757799	+	0.997125i	$0.475855\pi$
$938$	0	0
$939$	2.17170e7	0.803777
$940$	0	0
$941$	−1.08008e7	−0.397634	−0.198817	−	0.980037i	$-0.563710\pi$
$941$	−1.08008e7	−0.397634	−0.198817	+	0.980037i	$0.563710\pi$
$942$	0	0
$943$	9.49102e6	0.347563
$944$	0	0
$945$	9.15775e6	0.333587
$946$	0	0
$947$	−6.93307e6	−0.251218	−0.125609	−	0.992080i	$-0.540088\pi$
$947$	−6.93307e6	−0.251218	−0.125609	+	0.992080i	$0.540088\pi$
$948$	0	0
$949$	−3.76457e7	−1.35690
$950$	0	0
$951$	−9.27568e6	−0.332579
$952$	0	0
$953$	3.81395e7	1.36033	0.680163	−	0.733060i	$-0.261909\pi$
$953$	3.81395e7	1.36033	0.680163	+	0.733060i	$0.261909\pi$
$954$	0	0
$955$	−5.75084e6	−0.204043
$956$	0	0
$957$	−2.07721e6	−0.0733163
$958$	0	0
$959$	−2.20501e7	−0.774219
$960$	0	0
$961$	1.56184e7	0.545542
$962$	0	0
$963$	3.54382e6	0.123142
$964$	0	0
$965$	−1.78176e7	−0.615929
$966$	0	0
$967$	−1.52545e6	−0.0524603	−0.0262302	−	0.999656i	$-0.508350\pi$
$967$	−1.52545e6	−0.0524603	−0.0262302	+	0.999656i	$0.508350\pi$
$968$	0	0
$969$	6.32257e6	0.216314
$970$	0	0
$971$	4.37167e7	1.48799	0.743993	−	0.668187i	$-0.232929\pi$
$971$	4.37167e7	1.48799	0.743993	+	0.668187i	$0.232929\pi$
$972$	0	0
$973$	2.51553e7	0.851817
$974$	0	0
$975$	5.64261e7	1.90094
$976$	0	0
$977$	1.59223e6	0.0533666	0.0266833	−	0.999644i	$-0.491505\pi$
$977$	1.59223e6	0.0533666	0.0266833	+	0.999644i	$0.491505\pi$
$978$	0	0
$979$	1.18237e7	0.394271
$980$	0	0
$981$	1.53132e7	0.508035
$982$	0	0
$983$	−1.98366e7	−0.654762	−0.327381	−	0.944892i	$-0.606166\pi$
$983$	−1.98366e7	−0.654762	−0.327381	+	0.944892i	$0.606166\pi$
$984$	0	0
$985$	−5.24479e7	−1.72241
$986$	0	0
$987$	2.02810e7	0.662669
$988$	0	0
$989$	−5.66595e6	−0.184197
$990$	0	0
$991$	−5.77793e7	−1.86891	−0.934454	−	0.356083i	$-0.884112\pi$
$991$	−5.77793e7	−1.86891	−0.934454	+	0.356083i	$0.884112\pi$
$992$	0	0
$993$	2.08490e6	0.0670984
$994$	0	0
$995$	1.83753e7	0.588405
$996$	0	0
$997$	−2.48258e7	−0.790980	−0.395490	−	0.918470i	$-0.629425\pi$
$997$	−2.48258e7	−0.790980	−0.395490	+	0.918470i	$0.629425\pi$
$998$	0	0
$999$	5.87341e6	0.186199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.r.1.5		5
4.3	odd	2		69.6.a.e.1.1	✓	5
12.11	even	2		207.6.a.f.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.1	✓	5	4.3	odd	2
207.6.a.f.1.5		5	12.11	even	2
1104.6.a.r.1.5		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +99.7124 q^{5} -125.983 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +99.7124 q^{5} -125.983 q^{7} +81.0000 q^{9} -177.715 q^{11} -919.618 q^{13} -897.412 q^{15} +1568.24 q^{17} -447.960 q^{19} +1133.85 q^{21} +529.000 q^{23} +6817.57 q^{25} -729.000 q^{27} -1298.72 q^{29} +6651.88 q^{31} +1599.43 q^{33} -12562.1 q^{35} -8056.80 q^{37} +8276.56 q^{39} +17941.4 q^{41} -10710.7 q^{43} +8076.71 q^{45} +17886.9 q^{47} -935.293 q^{49} -14114.1 q^{51} -20343.9 q^{53} -17720.4 q^{55} +4031.64 q^{57} -28488.5 q^{59} +16222.3 q^{61} -10204.6 q^{63} -91697.4 q^{65} -54128.0 q^{67} -4761.00 q^{69} -42505.6 q^{71} +40936.2 q^{73} -61358.1 q^{75} +22389.1 q^{77} +24760.6 q^{79} +6561.00 q^{81} +30825.2 q^{83} +156373. q^{85} +11688.4 q^{87} -66531.6 q^{89} +115856. q^{91} -59866.9 q^{93} -44667.2 q^{95} -104506. q^{97} -14394.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) \) 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9	\( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) \) 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9 - 1100 * q^11 - 978 * q^13 - 846 * q^15 + 2522 * q^17 - 2060 * q^19 + 2448 * q^21 + 2645 * q^23 + 12035 * q^25 - 3645 * q^27 + 1526 * q^29 + 7392 * q^31 + 9900 * q^33 - 6056 * q^35 - 8210 * q^37 + 8802 * q^39 + 21250 * q^41 + 4548 * q^43 + 7614 * q^45 - 536 * q^47 - 27979 * q^49 - 22698 * q^51 - 11482 * q^53 + 77064 * q^55 + 18540 * q^57 - 74676 * q^59 - 44618 * q^61 - 22032 * q^63 - 24388 * q^65 + 1412 * q^67 - 23805 * q^69 - 37912 * q^71 + 46546 * q^73 - 108315 * q^75 + 157008 * q^77 - 50544 * q^79 + 32805 * q^81 - 89588 * q^83 + 147892 * q^85 - 13734 * q^87 + 280410 * q^89 + 27416 * q^91 - 66528 * q^93 - 203120 * q^95 + 90074 * q^97 - 89100 * q^99

Introduction

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