LMFDB - Embedded newform 882.6.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,Mod(1,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	882.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:	$141.458529075$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{4705}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1176 $ x^2 - x - 1176
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 294)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-33.7965$ of defining polynomial
Character	$\chi$	$=$	882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} +59.5930 q^{5} -64.0000 q^{8} +O(q^{10})$	$q-4.00000 q^{2} +16.0000 q^{4} +59.5930 q^{5} -64.0000 q^{8} -238.372 q^{10} +616.337 q^{11} +418.372 q^{13} +256.000 q^{16} -1793.90 q^{17} +1279.93 q^{19} +953.488 q^{20} -2465.35 q^{22} -4781.69 q^{23} +426.326 q^{25} -1673.49 q^{26} -1716.02 q^{29} +642.722 q^{31} -1024.00 q^{32} +7175.58 q^{34} -2360.02 q^{37} -5119.72 q^{38} -3813.95 q^{40} -15639.9 q^{41} -1638.61 q^{43} +9861.39 q^{44} +19126.7 q^{46} -20735.5 q^{47} -1705.30 q^{50} +6693.95 q^{52} +5347.93 q^{53} +36729.4 q^{55} +6864.09 q^{58} -16824.9 q^{59} -13527.2 q^{61} -2570.89 q^{62} +4096.00 q^{64} +24932.0 q^{65} -46483.5 q^{67} -28702.3 q^{68} +3273.91 q^{71} -76758.4 q^{73} +9440.09 q^{74} +20478.9 q^{76} -17731.4 q^{79} +15255.8 q^{80} +62559.5 q^{82} -78846.7 q^{83} -106904. q^{85} +6554.43 q^{86} -39445.6 q^{88} -74160.4 q^{89} -76507.0 q^{92} +82942.1 q^{94} +76274.9 q^{95} +24360.2 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} - 18 q^{5} - 128 q^{8}+O(q^{10}) $ 2 * q - 8 * q^2 + 32 * q^4 - 18 * q^5 - 128 * q^8	$ 2 q - 8 q^{2} + 32 q^{4} - 18 q^{5} - 128 q^{8} + 72 q^{10} - 2 q^{11} + 288 q^{13} + 512 q^{16} - 1530 q^{17} + 1188 q^{19} - 288 q^{20} + 8 q^{22} - 3390 q^{23} + 3322 q^{25} - 1152 q^{26} + 3976 q^{29} + 7596 q^{31} - 2048 q^{32} + 6120 q^{34} + 2688 q^{37} - 4752 q^{38} + 1152 q^{40} - 36630 q^{41} - 23032 q^{43} - 32 q^{44} + 13560 q^{46} - 864 q^{47} - 13288 q^{50} + 4608 q^{52} + 32920 q^{53} + 84708 q^{55} - 15904 q^{58} + 26712 q^{59} + 20412 q^{61} - 30384 q^{62} + 8192 q^{64} + 35048 q^{65} - 36172 q^{67} - 24480 q^{68} - 73706 q^{71} - 74772 q^{73} - 10752 q^{74} + 19008 q^{76} - 23116 q^{79} - 4608 q^{80} + 146520 q^{82} - 147816 q^{83} - 127380 q^{85} + 92128 q^{86} + 128 q^{88} - 164646 q^{89} - 54240 q^{92} + 3456 q^{94} + 83408 q^{95} + 162036 q^{97}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 - 18 * q^5 - 128 * q^8 + 72 * q^10 - 2 * q^11 + 288 * q^13 + 512 * q^16 - 1530 * q^17 + 1188 * q^19 - 288 * q^20 + 8 * q^22 - 3390 * q^23 + 3322 * q^25 - 1152 * q^26 + 3976 * q^29 + 7596 * q^31 - 2048 * q^32 + 6120 * q^34 + 2688 * q^37 - 4752 * q^38 + 1152 * q^40 - 36630 * q^41 - 23032 * q^43 - 32 * q^44 + 13560 * q^46 - 864 * q^47 - 13288 * q^50 + 4608 * q^52 + 32920 * q^53 + 84708 * q^55 - 15904 * q^58 + 26712 * q^59 + 20412 * q^61 - 30384 * q^62 + 8192 * q^64 + 35048 * q^65 - 36172 * q^67 - 24480 * q^68 - 73706 * q^71 - 74772 * q^73 - 10752 * q^74 + 19008 * q^76 - 23116 * q^79 - 4608 * q^80 + 146520 * q^82 - 147816 * q^83 - 127380 * q^85 + 92128 * q^86 + 128 * q^88 - 164646 * q^89 - 54240 * q^92 + 3456 * q^94 + 83408 * q^95 + 162036 * q^97

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	59.5930	1.06603	0.533016	−	0.846105i	$-0.321059\pi$
$5$	59.5930	1.06603	0.533016	+	0.846105i	$0.321059\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−238.372	−0.753798
$11$	616.337	1.53581	0.767903	−	0.640566i	$-0.221300\pi$
$11$	616.337	1.53581	0.767903	+	0.640566i	$0.221300\pi$
$12$	0	0
$13$	418.372	0.686601	0.343300	−	0.939226i	$-0.388455\pi$
$13$	418.372	0.686601	0.343300	+	0.939226i	$0.388455\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−1793.90	−1.50548	−0.752740	−	0.658318i	$-0.771268\pi$
$17$	−1793.90	−1.50548	−0.752740	+	0.658318i	$0.771268\pi$
$18$	0	0
$19$	1279.93	0.813396	0.406698	−	0.913563i	$-0.366680\pi$
$19$	1279.93	0.813396	0.406698	+	0.913563i	$0.366680\pi$
$20$	953.488	0.533016
$21$	0	0
$22$	−2465.35	−1.08598
$23$	−4781.69	−1.88478	−0.942392	−	0.334512i	$-0.891429\pi$
$23$	−4781.69	−1.88478	−0.942392	+	0.334512i	$0.891429\pi$
$24$	0	0
$25$	426.326	0.136424
$26$	−1673.49	−0.485500
$27$	0	0
$28$	0	0
$29$	−1716.02	−0.378903	−0.189451	−	0.981890i	$-0.560671\pi$
$29$	−1716.02	−0.378903	−0.189451	+	0.981890i	$0.560671\pi$
$30$	0	0
$31$	642.722	0.120121	0.0600605	−	0.998195i	$-0.480871\pi$
$31$	642.722	0.120121	0.0600605	+	0.998195i	$0.480871\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	7175.58	1.06453
$35$	0	0
$36$	0	0
$37$	−2360.02	−0.283408	−0.141704	−	0.989909i	$-0.545258\pi$
$37$	−2360.02	−0.283408	−0.141704	+	0.989909i	$0.545258\pi$
$38$	−5119.72	−0.575158
$39$	0	0
$40$	−3813.95	−0.376899
$41$	−15639.9	−1.45303	−0.726513	−	0.687152i	$-0.758860\pi$
$41$	−15639.9	−1.45303	−0.726513	+	0.687152i	$0.758860\pi$
$42$	0	0
$43$	−1638.61	−0.135146	−0.0675731	−	0.997714i	$-0.521526\pi$
$43$	−1638.61	−0.135146	−0.0675731	+	0.997714i	$0.521526\pi$
$44$	9861.39	0.767903
$45$	0	0
$46$	19126.7	1.33274
$47$	−20735.5	−1.36921	−0.684606	−	0.728914i	$-0.740026\pi$
$47$	−20735.5	−1.36921	−0.684606	+	0.728914i	$0.740026\pi$
$48$	0	0
$49$	0	0
$50$	−1705.30	−0.0964666
$51$	0	0
$52$	6693.95	0.343300
$53$	5347.93	0.261515	0.130757	−	0.991414i	$-0.458259\pi$
$53$	5347.93	0.261515	0.130757	+	0.991414i	$0.458259\pi$
$54$	0	0
$55$	36729.4	1.63722
$56$	0	0
$57$	0	0
$58$	6864.09	0.267925
$59$	−16824.9	−0.629250	−0.314625	−	0.949216i	$-0.601879\pi$
$59$	−16824.9	−0.629250	−0.314625	+	0.949216i	$0.601879\pi$
$60$	0	0
$61$	−13527.2	−0.465460	−0.232730	−	0.972541i	$-0.574766\pi$
$61$	−13527.2	−0.465460	−0.232730	+	0.972541i	$0.574766\pi$
$62$	−2570.89	−0.0849384
$63$	0	0
$64$	4096.00	0.125000
$65$	24932.0	0.731938
$66$	0	0
$67$	−46483.5	−1.26506	−0.632531	−	0.774535i	$-0.717984\pi$
$67$	−46483.5	−1.26506	−0.632531	+	0.774535i	$0.717984\pi$
$68$	−28702.3	−0.752740
$69$	0	0
$70$	0	0
$71$	3273.91	0.0770762	0.0385381	−	0.999257i	$-0.487730\pi$
$71$	3273.91	0.0770762	0.0385381	+	0.999257i	$0.487730\pi$
$72$	0	0
$73$	−76758.4	−1.68585	−0.842924	−	0.538032i	$-0.819168\pi$
$73$	−76758.4	−1.68585	−0.842924	+	0.538032i	$0.819168\pi$
$74$	9440.09	0.200400
$75$	0	0
$76$	20478.9	0.406698
$77$	0	0
$78$	0	0
$79$	−17731.4	−0.319650	−0.159825	−	0.987145i	$-0.551093\pi$
$79$	−17731.4	−0.319650	−0.159825	+	0.987145i	$0.551093\pi$
$80$	15255.8	0.266508
$81$	0	0
$82$	62559.5	1.02744
$83$	−78846.7	−1.25629	−0.628143	−	0.778098i	$-0.716185\pi$
$83$	−78846.7	−1.25629	−0.628143	+	0.778098i	$0.716185\pi$
$84$	0	0
$85$	−106904.	−1.60489
$86$	6554.43	0.0955628
$87$	0	0
$88$	−39445.6	−0.542990
$89$	−74160.4	−0.992424	−0.496212	−	0.868201i	$-0.665276\pi$
$89$	−74160.4	−0.992424	−0.496212	+	0.868201i	$0.665276\pi$
$90$	0	0
$91$	0	0
$92$	−76507.0	−0.942392
$93$	0	0
$94$	82942.1	0.968179
$95$	76274.9	0.867107
$96$	0	0
$97$	24360.2	0.262876	0.131438	−	0.991324i	$-0.458041\pi$
$97$	24360.2	0.262876	0.131438	+	0.991324i	$0.458041\pi$
$98$	0	0
$99$	0	0
$100$	6821.22	0.0682122
$101$	148780.	1.45124	0.725622	−	0.688093i	$-0.241552\pi$
$101$	148780.	1.45124	0.725622	+	0.688093i	$0.241552\pi$
$102$	0	0
$103$	205516.	1.90877	0.954383	−	0.298584i	$-0.0965143\pi$
$103$	205516.	1.90877	0.954383	+	0.298584i	$0.0965143\pi$
$104$	−26775.8	−0.242750
$105$	0	0
$106$	−21391.7	−0.184919
$107$	104578.	0.883038	0.441519	−	0.897252i	$-0.354440\pi$
$107$	104578.	0.883038	0.441519	+	0.897252i	$0.354440\pi$
$108$	0	0
$109$	212746.	1.71512	0.857560	−	0.514383i	$-0.171979\pi$
$109$	212746.	1.71512	0.857560	+	0.514383i	$0.171979\pi$
$110$	−146917.	−1.15769
$111$	0	0
$112$	0	0
$113$	−224886.	−1.65678	−0.828391	−	0.560150i	$-0.810744\pi$
$113$	−224886.	−1.65678	−0.828391	+	0.560150i	$0.810744\pi$
$114$	0	0
$115$	−284955.	−2.00924
$116$	−27456.4	−0.189451
$117$	0	0
$118$	67299.7	0.444947
$119$	0	0
$120$	0	0
$121$	218820.	1.35870
$122$	54108.7	0.329130
$123$	0	0
$124$	10283.6	0.0600605
$125$	−160822.	−0.920599
$126$	0	0
$127$	−44027.2	−0.242221	−0.121110	−	0.992639i	$-0.538646\pi$
$127$	−44027.2	−0.242221	−0.121110	+	0.992639i	$0.538646\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−99728.2	−0.517559
$131$	391132.	1.99134	0.995668	−	0.0929748i	$-0.0296376\pi$
$131$	391132.	1.99134	0.995668	+	0.0929748i	$0.0296376\pi$
$132$	0	0
$133$	0	0
$134$	185934.	0.894534
$135$	0	0
$136$	114809.	0.532267
$137$	−70982.2	−0.323108	−0.161554	−	0.986864i	$-0.551651\pi$
$137$	−70982.2	−0.323108	−0.161554	+	0.986864i	$0.551651\pi$
$138$	0	0
$139$	102764.	0.451133	0.225567	−	0.974228i	$-0.427577\pi$
$139$	102764.	0.451133	0.225567	+	0.974228i	$0.427577\pi$
$140$	0	0
$141$	0	0
$142$	−13095.6	−0.0545011
$143$	257858.	1.05449
$144$	0	0
$145$	−102263.	−0.403923
$146$	307034.	1.19208
$147$	0	0
$148$	−37760.4	−0.141704
$149$	226298.	0.835053	0.417527	−	0.908665i	$-0.362897\pi$
$149$	226298.	0.835053	0.417527	+	0.908665i	$0.362897\pi$
$150$	0	0
$151$	−548224.	−1.95666	−0.978330	−	0.207051i	$-0.933613\pi$
$151$	−548224.	−1.95666	−0.978330	+	0.207051i	$0.933613\pi$
$152$	−81915.5	−0.287579
$153$	0	0
$154$	0	0
$155$	38301.7	0.128053
$156$	0	0
$157$	241144.	0.780778	0.390389	−	0.920650i	$-0.372341\pi$
$157$	241144.	0.780778	0.390389	+	0.920650i	$0.372341\pi$
$158$	70925.5	0.226027
$159$	0	0
$160$	−61023.2	−0.188450
$161$	0	0
$162$	0	0
$163$	630902.	1.85991	0.929957	−	0.367668i	$-0.119844\pi$
$163$	630902.	1.85991	0.929957	+	0.367668i	$0.119844\pi$
$164$	−250238.	−0.726513
$165$	0	0
$166$	315387.	0.888328
$167$	−239113.	−0.663456	−0.331728	−	0.943375i	$-0.607632\pi$
$167$	−239113.	−0.663456	−0.331728	+	0.943375i	$0.607632\pi$
$168$	0	0
$169$	−196258.	−0.528579
$170$	427614.	1.13483
$171$	0	0
$172$	−26217.7	−0.0675731
$173$	302939.	0.769556	0.384778	−	0.923009i	$-0.374278\pi$
$173$	302939.	0.769556	0.384778	+	0.923009i	$0.374278\pi$
$174$	0	0
$175$	0	0
$176$	157782.	0.383952
$177$	0	0
$178$	296642.	0.701750
$179$	105990.	0.247247	0.123623	−	0.992329i	$-0.460549\pi$
$179$	105990.	0.247247	0.123623	+	0.992329i	$0.460549\pi$
$180$	0	0
$181$	−473727.	−1.07481	−0.537405	−	0.843324i	$-0.680595\pi$
$181$	−473727.	−1.07481	−0.537405	+	0.843324i	$0.680595\pi$
$182$	0	0
$183$	0	0
$184$	306028.	0.666371
$185$	−140641.	−0.302122
$186$	0	0
$187$	−1.10564e6	−2.31212
$188$	−331768.	−0.684606
$189$	0	0
$190$	−305099.	−0.613137
$191$	587813.	1.16589	0.582943	−	0.812513i	$-0.301901\pi$
$191$	587813.	1.16589	0.582943	+	0.812513i	$0.301901\pi$
$192$	0	0
$193$	−366861.	−0.708938	−0.354469	−	0.935068i	$-0.615338\pi$
$193$	−366861.	−0.708938	−0.354469	+	0.935068i	$0.615338\pi$
$194$	−97440.7	−0.185881
$195$	0	0
$196$	0	0
$197$	−727471.	−1.33552	−0.667760	−	0.744377i	$-0.732747\pi$
$197$	−727471.	−1.33552	−0.667760	+	0.744377i	$0.732747\pi$
$198$	0	0
$199$	−234812.	−0.420327	−0.210163	−	0.977666i	$-0.567400\pi$
$199$	−234812.	−0.420327	−0.210163	+	0.977666i	$0.567400\pi$
$200$	−27284.9	−0.0482333
$201$	0	0
$202$	−595119.	−1.02619
$203$	0	0
$204$	0	0
$205$	−932027.	−1.54897
$206$	−822065.	−1.34970
$207$	0	0
$208$	107103.	0.171650
$209$	788868.	1.24922
$210$	0	0
$211$	−308050.	−0.476337	−0.238169	−	0.971224i	$-0.576547\pi$
$211$	−308050.	−0.476337	−0.238169	+	0.971224i	$0.576547\pi$
$212$	85566.9	0.130757
$213$	0	0
$214$	−418311.	−0.624402
$215$	−97649.5	−0.144070
$216$	0	0
$217$	0	0
$218$	−850983.	−1.21277
$219$	0	0
$220$	587670.	0.818610
$221$	−750515.	−1.03366
$222$	0	0
$223$	510190.	0.687021	0.343510	−	0.939149i	$-0.388384\pi$
$223$	510190.	0.687021	0.343510	+	0.939149i	$0.388384\pi$
$224$	0	0
$225$	0	0
$226$	899542.	1.17152
$227$	−1.03803e6	−1.33705	−0.668523	−	0.743692i	$-0.733073\pi$
$227$	−1.03803e6	−1.33705	−0.668523	+	0.743692i	$0.733073\pi$
$228$	0	0
$229$	−1.03849e6	−1.30862	−0.654310	−	0.756226i	$-0.727041\pi$
$229$	−1.03849e6	−1.30862	−0.654310	+	0.756226i	$0.727041\pi$
$230$	1.13982e6	1.42075
$231$	0	0
$232$	109825.	0.133962
$233$	−1.31745e6	−1.58981	−0.794905	−	0.606734i	$-0.792479\pi$
$233$	−1.31745e6	−1.58981	−0.794905	+	0.606734i	$0.792479\pi$
$234$	0	0
$235$	−1.23569e6	−1.45962
$236$	−269199.	−0.314625
$237$	0	0
$238$	0	0
$239$	−800669.	−0.906689	−0.453345	−	0.891335i	$-0.649769\pi$
$239$	−800669.	−0.906689	−0.453345	+	0.891335i	$0.649769\pi$
$240$	0	0
$241$	402307.	0.446185	0.223093	−	0.974797i	$-0.428385\pi$
$241$	402307.	0.446185	0.223093	+	0.974797i	$0.428385\pi$
$242$	−875281.	−0.960747
$243$	0	0
$244$	−216435.	−0.232730
$245$	0	0
$246$	0	0
$247$	535487.	0.558479
$248$	−41134.2	−0.0424692
$249$	0	0
$250$	643288.	0.650962
$251$	1.29078e6	1.29320	0.646601	−	0.762828i	$-0.276190\pi$
$251$	1.29078e6	1.29320	0.646601	+	0.762828i	$0.276190\pi$
$252$	0	0
$253$	−2.94713e6	−2.89466
$254$	176109.	0.171276
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.76435e6	1.66629	0.833146	−	0.553053i	$-0.186537\pi$
$257$	1.76435e6	1.66629	0.833146	+	0.553053i	$0.186537\pi$
$258$	0	0
$259$	0	0
$260$	398913.	0.365969
$261$	0	0
$262$	−1.56453e6	−1.40809
$263$	−427745.	−0.381325	−0.190663	−	0.981656i	$-0.561064\pi$
$263$	−427745.	−0.381325	−0.190663	+	0.981656i	$0.561064\pi$
$264$	0	0
$265$	318699.	0.278783
$266$	0	0
$267$	0	0
$268$	−743736.	−0.632531
$269$	1.26691e6	1.06749	0.533744	−	0.845646i	$-0.320784\pi$
$269$	1.26691e6	1.06749	0.533744	+	0.845646i	$0.320784\pi$
$270$	0	0
$271$	−312034.	−0.258095	−0.129047	−	0.991638i	$-0.541192\pi$
$271$	−312034.	−0.258095	−0.129047	+	0.991638i	$0.541192\pi$
$272$	−459237.	−0.376370
$273$	0	0
$274$	283929.	0.228472
$275$	262760.	0.209521
$276$	0	0
$277$	−1.67860e6	−1.31446	−0.657229	−	0.753691i	$-0.728271\pi$
$277$	−1.67860e6	−1.31446	−0.657229	+	0.753691i	$0.728271\pi$
$278$	−411057.	−0.319000
$279$	0	0
$280$	0	0
$281$	765001.	0.577958	0.288979	−	0.957335i	$-0.406684\pi$
$281$	765001.	0.577958	0.288979	+	0.957335i	$0.406684\pi$
$282$	0	0
$283$	−85831.7	−0.0637062	−0.0318531	−	0.999493i	$-0.510141\pi$
$283$	−85831.7	−0.0637062	−0.0318531	+	0.999493i	$0.510141\pi$
$284$	52382.5	0.0385381
$285$	0	0
$286$	−1.03143e6	−0.745634
$287$	0	0
$288$	0	0
$289$	1.79820e6	1.26647
$290$	409052.	0.285616
$291$	0	0
$292$	−1.22813e6	−0.842924
$293$	97784.7	0.0665429	0.0332715	−	0.999446i	$-0.489407\pi$
$293$	97784.7	0.0665429	0.0332715	+	0.999446i	$0.489407\pi$
$294$	0	0
$295$	−1.00265e6	−0.670800
$296$	151041.	0.100200
$297$	0	0
$298$	−905190.	−0.590472
$299$	−2.00052e6	−1.29409
$300$	0	0
$301$	0	0
$302$	2.19289e6	1.38357
$303$	0	0
$304$	327662.	0.203349
$305$	−806125.	−0.496196
$306$	0	0
$307$	−1.75546e6	−1.06303	−0.531514	−	0.847050i	$-0.678377\pi$
$307$	−1.75546e6	−1.06303	−0.531514	+	0.847050i	$0.678377\pi$
$308$	0	0
$309$	0	0
$310$	−153207.	−0.0905470
$311$	1.80396e6	1.05761	0.528804	−	0.848744i	$-0.322641\pi$
$311$	1.80396e6	1.05761	0.528804	+	0.848744i	$0.322641\pi$
$312$	0	0
$313$	2.46429e6	1.42178	0.710888	−	0.703305i	$-0.248293\pi$
$313$	2.46429e6	1.42178	0.710888	+	0.703305i	$0.248293\pi$
$314$	−964576.	−0.552093
$315$	0	0
$316$	−283702.	−0.159825
$317$	−1.65286e6	−0.923822	−0.461911	−	0.886926i	$-0.652836\pi$
$317$	−1.65286e6	−0.923822	−0.461911	+	0.886926i	$0.652836\pi$
$318$	0	0
$319$	−1.05765e6	−0.581922
$320$	244093.	0.133254
$321$	0	0
$322$	0	0
$323$	−2.29606e6	−1.22455
$324$	0	0
$325$	178363.	0.0936690
$326$	−2.52361e6	−1.31516
$327$	0	0
$328$	1.00095e6	0.513722
$329$	0	0
$330$	0	0
$331$	−1.66814e6	−0.836881	−0.418440	−	0.908244i	$-0.637423\pi$
$331$	−1.66814e6	−0.836881	−0.418440	+	0.908244i	$0.637423\pi$
$332$	−1.26155e6	−0.628143
$333$	0	0
$334$	956452.	0.469134
$335$	−2.77009e6	−1.34860
$336$	0	0
$337$	−524769.	−0.251706	−0.125853	−	0.992049i	$-0.540167\pi$
$337$	−524769.	−0.251706	−0.125853	+	0.992049i	$0.540167\pi$
$338$	785031.	0.373762
$339$	0	0
$340$	−1.71046e6	−0.802444
$341$	396133.	0.184483
$342$	0	0
$343$	0	0
$344$	104871.	0.0477814
$345$	0	0
$346$	−1.21176e6	−0.544158
$347$	−1.92637e6	−0.858846	−0.429423	−	0.903103i	$-0.641283\pi$
$347$	−1.92637e6	−0.858846	−0.429423	+	0.903103i	$0.641283\pi$
$348$	0	0
$349$	340342.	0.149573	0.0747864	−	0.997200i	$-0.476173\pi$
$349$	340342.	0.149573	0.0747864	+	0.997200i	$0.476173\pi$
$350$	0	0
$351$	0	0
$352$	−631129.	−0.271495
$353$	74791.2	0.0319458	0.0159729	−	0.999872i	$-0.494915\pi$
$353$	74791.2	0.0319458	0.0159729	+	0.999872i	$0.494915\pi$
$354$	0	0
$355$	195102.	0.0821657
$356$	−1.18657e6	−0.496212
$357$	0	0
$358$	−423958.	−0.174830
$359$	−2.41754e6	−0.990004	−0.495002	−	0.868892i	$-0.664833\pi$
$359$	−2.41754e6	−0.990004	−0.495002	+	0.868892i	$0.664833\pi$
$360$	0	0
$361$	−837878.	−0.338386
$362$	1.89491e6	0.760005
$363$	0	0
$364$	0	0
$365$	−4.57426e6	−1.79717
$366$	0	0
$367$	161864.	0.0627314	0.0313657	−	0.999508i	$-0.490014\pi$
$367$	161864.	0.0627314	0.0313657	+	0.999508i	$0.490014\pi$
$368$	−1.22411e6	−0.471196
$369$	0	0
$370$	562563.	0.213632
$371$	0	0
$372$	0	0
$373$	77002.5	0.0286571	0.0143286	−	0.999897i	$-0.495439\pi$
$373$	77002.5	0.0286571	0.0143286	+	0.999897i	$0.495439\pi$
$374$	4.42258e6	1.63492
$375$	0	0
$376$	1.32707e6	0.484089
$377$	−717936.	−0.260155
$378$	0	0
$379$	−774165.	−0.276844	−0.138422	−	0.990373i	$-0.544203\pi$
$379$	−774165.	−0.276844	−0.138422	+	0.990373i	$0.544203\pi$
$380$	1.22040e6	0.433553
$381$	0	0
$382$	−2.35125e6	−0.824406
$383$	−3.63328e6	−1.26561	−0.632807	−	0.774309i	$-0.718098\pi$
$383$	−3.63328e6	−1.26561	−0.632807	+	0.774309i	$0.718098\pi$
$384$	0	0
$385$	0	0
$386$	1.46744e6	0.501295
$387$	0	0
$388$	389763.	0.131438
$389$	2.51908e6	0.844048	0.422024	−	0.906585i	$-0.361320\pi$
$389$	2.51908e6	0.844048	0.422024	+	0.906585i	$0.361320\pi$
$390$	0	0
$391$	8.57784e6	2.83750
$392$	0	0
$393$	0	0
$394$	2.90989e6	0.944355
$395$	−1.05667e6	−0.340757
$396$	0	0
$397$	5.59278e6	1.78095	0.890474	−	0.455034i	$-0.150373\pi$
$397$	5.59278e6	1.78095	0.890474	+	0.455034i	$0.150373\pi$
$398$	939246.	0.297216
$399$	0	0
$400$	109139.	0.0341061
$401$	191062.	0.0593353	0.0296676	−	0.999560i	$-0.490555\pi$
$401$	191062.	0.0593353	0.0296676	+	0.999560i	$0.490555\pi$
$402$	0	0
$403$	268897.	0.0824751
$404$	2.38048e6	0.725622
$405$	0	0
$406$	0	0
$407$	−1.45457e6	−0.435260
$408$	0	0
$409$	4.45805e6	1.31776	0.658880	−	0.752248i	$-0.271030\pi$
$409$	4.45805e6	1.31776	0.658880	+	0.752248i	$0.271030\pi$
$410$	3.72811e6	1.09529
$411$	0	0
$412$	3.28826e6	0.954383
$413$	0	0
$414$	0	0
$415$	−4.69871e6	−1.33924
$416$	−428413.	−0.121375
$417$	0	0
$418$	−3.15547e6	−0.883332
$419$	−4.76120e6	−1.32489	−0.662447	−	0.749109i	$-0.730482\pi$
$419$	−4.76120e6	−1.32489	−0.662447	+	0.749109i	$0.730482\pi$
$420$	0	0
$421$	−1.38498e6	−0.380835	−0.190418	−	0.981703i	$-0.560984\pi$
$421$	−1.38498e6	−0.380835	−0.190418	+	0.981703i	$0.560984\pi$
$422$	1.23220e6	0.336821
$423$	0	0
$424$	−342268.	−0.0924595
$425$	−764784.	−0.205384
$426$	0	0
$427$	0	0
$428$	1.67324e6	0.441519
$429$	0	0
$430$	390598.	0.101873
$431$	−4.55891e6	−1.18214	−0.591069	−	0.806621i	$-0.701294\pi$
$431$	−4.55891e6	−1.18214	−0.591069	+	0.806621i	$0.701294\pi$
$432$	0	0
$433$	−6.00870e6	−1.54014	−0.770071	−	0.637958i	$-0.779779\pi$
$433$	−6.00870e6	−1.54014	−0.770071	+	0.637958i	$0.779779\pi$
$434$	0	0
$435$	0	0
$436$	3.40393e6	0.857560
$437$	−6.12022e6	−1.53308
$438$	0	0
$439$	560503.	0.138809	0.0694044	−	0.997589i	$-0.477890\pi$
$439$	560503.	0.138809	0.0694044	+	0.997589i	$0.477890\pi$
$440$	−2.35068e6	−0.578844
$441$	0	0
$442$	3.00206e6	0.730910
$443$	−3.34074e6	−0.808786	−0.404393	−	0.914585i	$-0.632517\pi$
$443$	−3.34074e6	−0.808786	−0.404393	+	0.914585i	$0.632517\pi$
$444$	0	0
$445$	−4.41944e6	−1.05796
$446$	−2.04076e6	−0.485797
$447$	0	0
$448$	0	0
$449$	6.47842e6	1.51654	0.758269	−	0.651942i	$-0.226045\pi$
$449$	6.47842e6	1.51654	0.758269	+	0.651942i	$0.226045\pi$
$450$	0	0
$451$	−9.63943e6	−2.23157
$452$	−3.59817e6	−0.828391
$453$	0	0
$454$	4.15213e6	0.945434
$455$	0	0
$456$	0	0
$457$	4.31138e6	0.965663	0.482831	−	0.875713i	$-0.339608\pi$
$457$	4.31138e6	0.965663	0.482831	+	0.875713i	$0.339608\pi$
$458$	4.15396e6	0.925334
$459$	0	0
$460$	−4.55928e6	−1.00462
$461$	320951.	0.0703374	0.0351687	−	0.999381i	$-0.488803\pi$
$461$	320951.	0.0703374	0.0351687	+	0.999381i	$0.488803\pi$
$462$	0	0
$463$	−3.14239e6	−0.681251	−0.340626	−	0.940199i	$-0.610639\pi$
$463$	−3.14239e6	−0.681251	−0.340626	+	0.940199i	$0.610639\pi$
$464$	−439302.	−0.0947257
$465$	0	0
$466$	5.26981e6	1.12417
$467$	−4.14587e6	−0.879677	−0.439839	−	0.898077i	$-0.644964\pi$
$467$	−4.14587e6	−0.879677	−0.439839	+	0.898077i	$0.644964\pi$
$468$	0	0
$469$	0	0
$470$	4.94277e6	1.03211
$471$	0	0
$472$	1.07679e6	0.222473
$473$	−1.00993e6	−0.207558
$474$	0	0
$475$	545667.	0.110967
$476$	0	0
$477$	0	0
$478$	3.20268e6	0.641126
$479$	257678.	0.0513142	0.0256571	−	0.999671i	$-0.491832\pi$
$479$	257678.	0.0513142	0.0256571	+	0.999671i	$0.491832\pi$
$480$	0	0
$481$	−987367.	−0.194588
$482$	−1.60923e6	−0.315501
$483$	0	0
$484$	3.50113e6	0.679351
$485$	1.45170e6	0.280234
$486$	0	0
$487$	1.72033e6	0.328692	0.164346	−	0.986403i	$-0.447449\pi$
$487$	1.72033e6	0.328692	0.164346	+	0.986403i	$0.447449\pi$
$488$	865739.	0.164565
$489$	0	0
$490$	0	0
$491$	−6.49101e6	−1.21509	−0.607546	−	0.794285i	$-0.707846\pi$
$491$	−6.49101e6	−1.21509	−0.607546	+	0.794285i	$0.707846\pi$
$492$	0	0
$493$	3.07836e6	0.570430
$494$	−2.14195e6	−0.394904
$495$	0	0
$496$	164537.	0.0300302
$497$	0	0
$498$	0	0
$499$	379631.	0.0682512	0.0341256	−	0.999418i	$-0.489135\pi$
$499$	379631.	0.0682512	0.0341256	+	0.999418i	$0.489135\pi$
$500$	−2.57315e6	−0.460300
$501$	0	0
$502$	−5.16311e6	−0.914433
$503$	8.60340e6	1.51618	0.758089	−	0.652151i	$-0.226133\pi$
$503$	8.60340e6	1.51618	0.758089	+	0.652151i	$0.226133\pi$
$504$	0	0
$505$	8.86624e6	1.54707
$506$	1.17885e7	2.04684
$507$	0	0
$508$	−704435.	−0.121110
$509$	−1.03543e6	−0.177144	−0.0885718	−	0.996070i	$-0.528230\pi$
$509$	−1.03543e6	−0.177144	−0.0885718	+	0.996070i	$0.528230\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−7.05739e6	−1.17825
$515$	1.22473e7	2.03481
$516$	0	0
$517$	−1.27801e7	−2.10284
$518$	0	0
$519$	0	0
$520$	−1.59565e6	−0.258779
$521$	−6.42373e6	−1.03680	−0.518398	−	0.855140i	$-0.673471\pi$
$521$	−6.42373e6	−1.03680	−0.518398	+	0.855140i	$0.673471\pi$
$522$	0	0
$523$	8.06384e6	1.28910	0.644551	−	0.764561i	$-0.277044\pi$
$523$	8.06384e6	1.28910	0.644551	+	0.764561i	$0.277044\pi$
$524$	6.25811e6	0.995668
$525$	0	0
$526$	1.71098e6	0.269638
$527$	−1.15298e6	−0.180840
$528$	0	0
$529$	1.64282e7	2.55241
$530$	−1.27480e6	−0.197129
$531$	0	0
$532$	0	0
$533$	−6.54329e6	−0.997649
$534$	0	0
$535$	6.23210e6	0.941347
$536$	2.97494e6	0.447267
$537$	0	0
$538$	−5.06762e6	−0.754829
$539$	0	0
$540$	0	0
$541$	−8.62213e6	−1.26655	−0.633274	−	0.773928i	$-0.718289\pi$
$541$	−8.62213e6	−1.26655	−0.633274	+	0.773928i	$0.718289\pi$
$542$	1.24814e6	0.182500
$543$	0	0
$544$	1.83695e6	0.266134
$545$	1.26782e7	1.82837
$546$	0	0
$547$	−3.51266e6	−0.501958	−0.250979	−	0.967993i	$-0.580753\pi$
$547$	−3.51266e6	−0.501958	−0.250979	+	0.967993i	$0.580753\pi$
$548$	−1.13571e6	−0.161554
$549$	0	0
$550$	−1.05104e6	−0.148154
$551$	−2.19639e6	−0.308198
$552$	0	0
$553$	0	0
$554$	6.71438e6	0.929462
$555$	0	0
$556$	1.64423e6	0.225567
$557$	1.22183e7	1.66868	0.834340	−	0.551251i	$-0.185849\pi$
$557$	1.22183e7	1.66868	0.834340	+	0.551251i	$0.185849\pi$
$558$	0	0
$559$	−685548.	−0.0927915
$560$	0	0
$561$	0	0
$562$	−3.06000e6	−0.408678
$563$	−1.48761e6	−0.197796	−0.0988980	−	0.995098i	$-0.531532\pi$
$563$	−1.48761e6	−0.197796	−0.0988980	+	0.995098i	$0.531532\pi$
$564$	0	0
$565$	−1.34016e7	−1.76618
$566$	343327.	0.0450471
$567$	0	0
$568$	−209530.	−0.0272506
$569$	−5.83501e6	−0.755547	−0.377773	−	0.925898i	$-0.623310\pi$
$569$	−5.83501e6	−0.755547	−0.377773	+	0.925898i	$0.623310\pi$
$570$	0	0
$571$	9.83227e6	1.26201	0.631006	−	0.775778i	$-0.282642\pi$
$571$	9.83227e6	1.26201	0.631006	+	0.775778i	$0.282642\pi$
$572$	4.12573e6	0.527243
$573$	0	0
$574$	0	0
$575$	−2.03856e6	−0.257130
$576$	0	0
$577$	324006.	0.0405147	0.0202574	−	0.999795i	$-0.493551\pi$
$577$	324006.	0.0405147	0.0202574	+	0.999795i	$0.493551\pi$
$578$	−7.19281e6	−0.895528
$579$	0	0
$580$	−1.63621e6	−0.201961
$581$	0	0
$582$	0	0
$583$	3.29613e6	0.401636
$584$	4.91254e6	0.596038
$585$	0	0
$586$	−391139.	−0.0470530
$587$	−4.27646e6	−0.512258	−0.256129	−	0.966643i	$-0.582447\pi$
$587$	−4.27646e6	−0.512258	−0.256129	+	0.966643i	$0.582447\pi$
$588$	0	0
$589$	822639.	0.0977060
$590$	4.01059e6	0.474327
$591$	0	0
$592$	−604166.	−0.0708519
$593$	1.58930e7	1.85596	0.927982	−	0.372625i	$-0.121542\pi$
$593$	1.58930e7	1.85596	0.927982	+	0.372625i	$0.121542\pi$
$594$	0	0
$595$	0	0
$596$	3.62076e6	0.417527
$597$	0	0
$598$	8.00209e6	0.915062
$599$	1.75061e7	1.99353	0.996766	−	0.0803623i	$-0.0256077\pi$
$599$	1.75061e7	1.99353	0.996766	+	0.0803623i	$0.0256077\pi$
$600$	0	0
$601$	−5.98914e6	−0.676361	−0.338180	−	0.941081i	$-0.609811\pi$
$601$	−5.98914e6	−0.676361	−0.338180	+	0.941081i	$0.609811\pi$
$602$	0	0
$603$	0	0
$604$	−8.77158e6	−0.978330
$605$	1.30402e7	1.44842
$606$	0	0
$607$	−3.27501e6	−0.360778	−0.180389	−	0.983595i	$-0.557736\pi$
$607$	−3.27501e6	−0.360778	−0.180389	+	0.983595i	$0.557736\pi$
$608$	−1.31065e6	−0.143790
$609$	0	0
$610$	3.22450e6	0.350863
$611$	−8.67516e6	−0.940101
$612$	0	0
$613$	−7.62114e6	−0.819160	−0.409580	−	0.912274i	$-0.634325\pi$
$613$	−7.62114e6	−0.819160	−0.409580	+	0.912274i	$0.634325\pi$
$614$	7.02183e6	0.751674
$615$	0	0
$616$	0	0
$617$	−1.28554e7	−1.35948	−0.679741	−	0.733453i	$-0.737908\pi$
$617$	−1.28554e7	−1.35948	−0.679741	+	0.733453i	$0.737908\pi$
$618$	0	0
$619$	−8.52350e6	−0.894111	−0.447055	−	0.894506i	$-0.647527\pi$
$619$	−8.52350e6	−0.894111	−0.447055	+	0.894506i	$0.647527\pi$
$620$	612828.	0.0640264
$621$	0	0
$622$	−7.21582e6	−0.747842
$623$	0	0
$624$	0	0
$625$	−1.09161e7	−1.11781
$626$	−9.85716e6	−1.00535
$627$	0	0
$628$	3.85831e6	0.390389
$629$	4.23363e6	0.426664
$630$	0	0
$631$	−1.37373e7	−1.37350	−0.686750	−	0.726894i	$-0.740963\pi$
$631$	−1.37373e7	−1.37350	−0.686750	+	0.726894i	$0.740963\pi$
$632$	1.13481e6	0.113013
$633$	0	0
$634$	6.61144e6	0.653241
$635$	−2.62371e6	−0.258215
$636$	0	0
$637$	0	0
$638$	4.23059e6	0.411481
$639$	0	0
$640$	−976372.	−0.0942248
$641$	5.04466e6	0.484939	0.242470	−	0.970159i	$-0.422043\pi$
$641$	5.04466e6	0.484939	0.242470	+	0.970159i	$0.422043\pi$
$642$	0	0
$643$	9.94759e6	0.948835	0.474417	−	0.880300i	$-0.342659\pi$
$643$	9.94759e6	0.948835	0.474417	+	0.880300i	$0.342659\pi$
$644$	0	0
$645$	0	0
$646$	9.18424e6	0.865888
$647$	−1.26276e6	−0.118593	−0.0592966	−	0.998240i	$-0.518886\pi$
$647$	−1.26276e6	−0.118593	−0.0592966	+	0.998240i	$0.518886\pi$
$648$	0	0
$649$	−1.03698e7	−0.966406
$650$	−713451.	−0.0662340
$651$	0	0
$652$	1.00944e7	0.929957
$653$	1.28147e7	1.17605	0.588026	−	0.808842i	$-0.299905\pi$
$653$	1.28147e7	1.17605	0.588026	+	0.808842i	$0.299905\pi$
$654$	0	0
$655$	2.33087e7	2.12283
$656$	−4.00381e6	−0.363257
$657$	0	0
$658$	0	0
$659$	3.81355e6	0.342071	0.171035	−	0.985265i	$-0.445289\pi$
$659$	3.81355e6	0.342071	0.171035	+	0.985265i	$0.445289\pi$
$660$	0	0
$661$	1.66244e7	1.47994	0.739968	−	0.672642i	$-0.234841\pi$
$661$	1.66244e7	1.47994	0.739968	+	0.672642i	$0.234841\pi$
$662$	6.67257e6	0.591764
$663$	0	0
$664$	5.04619e6	0.444164
$665$	0	0
$666$	0	0
$667$	8.20548e6	0.714150
$668$	−3.82581e6	−0.331728
$669$	0	0
$670$	1.10804e7	0.953602
$671$	−8.33730e6	−0.714857
$672$	0	0
$673$	1.76139e7	1.49905	0.749527	−	0.661974i	$-0.230281\pi$
$673$	1.76139e7	1.49905	0.749527	+	0.661974i	$0.230281\pi$
$674$	2.09908e6	0.177983
$675$	0	0
$676$	−3.14013e6	−0.264290
$677$	1.78430e7	1.49622	0.748112	−	0.663573i	$-0.230961\pi$
$677$	1.78430e7	1.49622	0.748112	+	0.663573i	$0.230961\pi$
$678$	0	0
$679$	0	0
$680$	6.84183e6	0.567414
$681$	0	0
$682$	−1.58453e6	−0.130449
$683$	−1.41532e7	−1.16092	−0.580459	−	0.814289i	$-0.697127\pi$
$683$	−1.41532e7	−1.16092	−0.580459	+	0.814289i	$0.697127\pi$
$684$	0	0
$685$	−4.23004e6	−0.344444
$686$	0	0
$687$	0	0
$688$	−419484.	−0.0337865
$689$	2.23743e6	0.179556
$690$	0	0
$691$	2.22535e6	0.177297	0.0886487	−	0.996063i	$-0.471745\pi$
$691$	2.22535e6	0.177297	0.0886487	+	0.996063i	$0.471745\pi$
$692$	4.84703e6	0.384778
$693$	0	0
$694$	7.70547e6	0.607296
$695$	6.12403e6	0.480923
$696$	0	0
$697$	2.80563e7	2.18750
$698$	−1.36137e6	−0.105764
$699$	0	0
$700$	0	0
$701$	4.49378e6	0.345396	0.172698	−	0.984975i	$-0.444752\pi$
$701$	4.49378e6	0.345396	0.172698	+	0.984975i	$0.444752\pi$
$702$	0	0
$703$	−3.02066e6	−0.230523
$704$	2.52452e6	0.191976
$705$	0	0
$706$	−299165.	−0.0225891
$707$	0	0
$708$	0	0
$709$	−1.54232e7	−1.15228	−0.576141	−	0.817350i	$-0.695442\pi$
$709$	−1.54232e7	−1.15228	−0.576141	+	0.817350i	$0.695442\pi$
$710$	−780408.	−0.0580999
$711$	0	0
$712$	4.74627e6	0.350875
$713$	−3.07329e6	−0.226402
$714$	0	0
$715$	1.53665e7	1.12412
$716$	1.69583e6	0.123623
$717$	0	0
$718$	9.67014e6	0.700038
$719$	1.07276e7	0.773890	0.386945	−	0.922103i	$-0.373530\pi$
$719$	1.07276e7	0.773890	0.386945	+	0.922103i	$0.373530\pi$
$720$	0	0
$721$	0	0
$722$	3.35151e6	0.239275
$723$	0	0
$724$	−7.57963e6	−0.537405
$725$	−731585.	−0.0516916
$726$	0	0
$727$	2.05652e7	1.44310	0.721551	−	0.692362i	$-0.243430\pi$
$727$	2.05652e7	1.44310	0.721551	+	0.692362i	$0.243430\pi$
$728$	0	0
$729$	0	0
$730$	1.82971e7	1.27079
$731$	2.93949e6	0.203460
$732$	0	0
$733$	3.29322e6	0.226392	0.113196	−	0.993573i	$-0.463891\pi$
$733$	3.29322e6	0.226392	0.113196	+	0.993573i	$0.463891\pi$
$734$	−647455.	−0.0443578
$735$	0	0
$736$	4.89645e6	0.333186
$737$	−2.86495e7	−1.94289
$738$	0	0
$739$	−7.71767e6	−0.519847	−0.259923	−	0.965629i	$-0.583697\pi$
$739$	−7.71767e6	−0.519847	−0.259923	+	0.965629i	$0.583697\pi$
$740$	−2.25025e6	−0.151061
$741$	0	0
$742$	0	0
$743$	8.33553e6	0.553938	0.276969	−	0.960879i	$-0.410670\pi$
$743$	8.33553e6	0.553938	0.276969	+	0.960879i	$0.410670\pi$
$744$	0	0
$745$	1.34857e7	0.890193
$746$	−308010.	−0.0202636
$747$	0	0
$748$	−1.76903e7	−1.15606
$749$	0	0
$750$	0	0
$751$	−1.80694e7	−1.16908	−0.584539	−	0.811366i	$-0.698725\pi$
$751$	−1.80694e7	−1.16908	−0.584539	+	0.811366i	$0.698725\pi$
$752$	−5.30830e6	−0.342303
$753$	0	0
$754$	2.87174e6	0.183957
$755$	−3.26703e7	−2.08586
$756$	0	0
$757$	−8.13311e6	−0.515842	−0.257921	−	0.966166i	$-0.583037\pi$
$757$	−8.13311e6	−0.515842	−0.257921	+	0.966166i	$0.583037\pi$
$758$	3.09666e6	0.195758
$759$	0	0
$760$	−4.88159e6	−0.306568
$761$	−2.01691e6	−0.126248	−0.0631239	−	0.998006i	$-0.520106\pi$
$761$	−2.01691e6	−0.126248	−0.0631239	+	0.998006i	$0.520106\pi$
$762$	0	0
$763$	0	0
$764$	9.40501e6	0.582943
$765$	0	0
$766$	1.45331e7	0.894925
$767$	−7.03908e6	−0.432043
$768$	0	0
$769$	1.18971e7	0.725478	0.362739	−	0.931891i	$-0.381842\pi$
$769$	1.18971e7	0.725478	0.362739	+	0.931891i	$0.381842\pi$
$770$	0	0
$771$	0	0
$772$	−5.86977e6	−0.354469
$773$	−172292.	−0.0103709	−0.00518545	−	0.999987i	$-0.501651\pi$
$773$	−172292.	−0.0103709	−0.00518545	+	0.999987i	$0.501651\pi$
$774$	0	0
$775$	274009.	0.0163874
$776$	−1.55905e6	−0.0929407
$777$	0	0
$778$	−1.00763e7	−0.596832
$779$	−2.00179e7	−1.18189
$780$	0	0
$781$	2.01783e6	0.118374
$782$	−3.43114e7	−2.00642
$783$	0	0
$784$	0	0
$785$	1.43705e7	0.832334
$786$	0	0
$787$	−1.09131e7	−0.628075	−0.314037	−	0.949411i	$-0.601682\pi$
$787$	−1.09131e7	−0.628075	−0.314037	+	0.949411i	$0.601682\pi$
$788$	−1.16395e7	−0.667760
$789$	0	0
$790$	4.22666e6	0.240952
$791$	0	0
$792$	0	0
$793$	−5.65939e6	−0.319585
$794$	−2.23711e7	−1.25932
$795$	0	0
$796$	−3.75699e6	−0.210163
$797$	2.45403e7	1.36847	0.684234	−	0.729263i	$-0.260137\pi$
$797$	2.45403e7	1.36847	0.684234	+	0.729263i	$0.260137\pi$
$798$	0	0
$799$	3.71974e7	2.06132
$800$	−436558.	−0.0241166
$801$	0	0
$802$	−764247.	−0.0419564
$803$	−4.73090e7	−2.58914
$804$	0	0
$805$	0	0
$806$	−1.07559e6	−0.0583187
$807$	0	0
$808$	−9.52191e6	−0.513093
$809$	3.07074e7	1.64958	0.824788	−	0.565442i	$-0.191294\pi$
$809$	3.07074e7	1.64958	0.824788	+	0.565442i	$0.191294\pi$
$810$	0	0
$811$	1.51675e7	0.809771	0.404886	−	0.914367i	$-0.367311\pi$
$811$	1.51675e7	0.809771	0.404886	+	0.914367i	$0.367311\pi$
$812$	0	0
$813$	0	0
$814$	5.81828e6	0.307775
$815$	3.75974e7	1.98273
$816$	0	0
$817$	−2.09730e6	−0.109927
$818$	−1.78322e7	−0.931797
$819$	0	0
$820$	−1.49124e7	−0.774486
$821$	−3.48574e6	−0.180483	−0.0902416	−	0.995920i	$-0.528764\pi$
$821$	−3.48574e6	−0.180483	−0.0902416	+	0.995920i	$0.528764\pi$
$822$	0	0
$823$	−2.20822e7	−1.13643	−0.568214	−	0.822881i	$-0.692365\pi$
$823$	−2.20822e7	−1.13643	−0.568214	+	0.822881i	$0.692365\pi$
$824$	−1.31530e7	−0.674851
$825$	0	0
$826$	0	0
$827$	−2.28055e7	−1.15951	−0.579757	−	0.814789i	$-0.696853\pi$
$827$	−2.28055e7	−1.15951	−0.579757	+	0.814789i	$0.696853\pi$
$828$	0	0
$829$	−3.56532e7	−1.80182	−0.900912	−	0.434002i	$-0.857101\pi$
$829$	−3.56532e7	−1.80182	−0.900912	+	0.434002i	$0.857101\pi$
$830$	1.87948e7	0.946986
$831$	0	0
$832$	1.71365e6	0.0858251
$833$	0	0
$834$	0	0
$835$	−1.42495e7	−0.707266
$836$	1.26219e7	0.624610
$837$	0	0
$838$	1.90448e7	0.936842
$839$	−2.55545e7	−1.25332	−0.626662	−	0.779291i	$-0.715579\pi$
$839$	−2.55545e7	−1.25332	−0.626662	+	0.779291i	$0.715579\pi$
$840$	0	0
$841$	−1.75664e7	−0.856433
$842$	5.53990e6	0.269291
$843$	0	0
$844$	−4.92879e6	−0.238169
$845$	−1.16956e7	−0.563483
$846$	0	0
$847$	0	0
$848$	1.36907e6	0.0653787
$849$	0	0
$850$	3.05914e6	0.145228
$851$	1.12849e7	0.534162
$852$	0	0
$853$	2.55946e6	0.120441	0.0602206	−	0.998185i	$-0.480820\pi$
$853$	2.55946e6	0.120441	0.0602206	+	0.998185i	$0.480820\pi$
$854$	0	0
$855$	0	0
$856$	−6.69297e6	−0.312201
$857$	−4.96836e6	−0.231079	−0.115540	−	0.993303i	$-0.536860\pi$
$857$	−4.96836e6	−0.231079	−0.115540	+	0.993303i	$0.536860\pi$
$858$	0	0
$859$	−1.11897e7	−0.517411	−0.258706	−	0.965956i	$-0.583296\pi$
$859$	−1.11897e7	−0.517411	−0.258706	+	0.965956i	$0.583296\pi$
$860$	−1.56239e6	−0.0720351
$861$	0	0
$862$	1.82356e7	0.835897
$863$	2.97620e6	0.136030	0.0680151	−	0.997684i	$-0.478333\pi$
$863$	2.97620e6	0.136030	0.0680151	+	0.997684i	$0.478333\pi$
$864$	0	0
$865$	1.80530e7	0.820371
$866$	2.40348e7	1.08905
$867$	0	0
$868$	0	0
$869$	−1.09285e7	−0.490920
$870$	0	0
$871$	−1.94474e7	−0.868593
$872$	−1.36157e7	−0.606387
$873$	0	0
$874$	2.44809e7	1.08405
$875$	0	0
$876$	0	0
$877$	2.84689e7	1.24989	0.624945	−	0.780669i	$-0.285121\pi$
$877$	2.84689e7	1.24989	0.624945	+	0.780669i	$0.285121\pi$
$878$	−2.24201e6	−0.0981526
$879$	0	0
$880$	9.40272e6	0.409305
$881$	1.97744e7	0.858346	0.429173	−	0.903222i	$-0.358805\pi$
$881$	1.97744e7	0.858346	0.429173	+	0.903222i	$0.358805\pi$
$882$	0	0
$883$	1.88722e7	0.814556	0.407278	−	0.913304i	$-0.366478\pi$
$883$	1.88722e7	0.814556	0.407278	+	0.913304i	$0.366478\pi$
$884$	−1.20082e7	−0.516831
$885$	0	0
$886$	1.33630e7	0.571898
$887$	1.71526e7	0.732016	0.366008	−	0.930612i	$-0.380724\pi$
$887$	1.71526e7	0.732016	0.366008	+	0.930612i	$0.380724\pi$
$888$	0	0
$889$	0	0
$890$	1.76778e7	0.748088
$891$	0	0
$892$	8.16304e6	0.343510
$893$	−2.65400e7	−1.11371
$894$	0	0
$895$	6.31624e6	0.263573
$896$	0	0
$897$	0	0
$898$	−2.59137e7	−1.07235
$899$	−1.10292e6	−0.0455142
$900$	0	0
$901$	−9.59363e6	−0.393705
$902$	3.85577e7	1.57796
$903$	0	0
$904$	1.43927e7	0.585761
$905$	−2.82308e7	−1.14578
$906$	0	0
$907$	−2.02760e7	−0.818399	−0.409199	−	0.912445i	$-0.634192\pi$
$907$	−2.02760e7	−0.818399	−0.409199	+	0.912445i	$0.634192\pi$
$908$	−1.66085e7	−0.668523
$909$	0	0
$910$	0	0
$911$	1.53633e7	0.613322	0.306661	−	0.951819i	$-0.400788\pi$
$911$	1.53633e7	0.613322	0.306661	+	0.951819i	$0.400788\pi$
$912$	0	0
$913$	−4.85961e7	−1.92941
$914$	−1.72455e7	−0.682827
$915$	0	0
$916$	−1.66158e7	−0.654310
$917$	0	0
$918$	0	0
$919$	2.10973e7	0.824021	0.412010	−	0.911179i	$-0.364827\pi$
$919$	2.10973e7	0.824021	0.412010	+	0.911179i	$0.364827\pi$
$920$	1.82371e7	0.710373
$921$	0	0
$922$	−1.28380e6	−0.0497360
$923$	1.36971e6	0.0529206
$924$	0	0
$925$	−1.00614e6	−0.0386637
$926$	1.25695e7	0.481717
$927$	0	0
$928$	1.75721e6	0.0669812
$929$	1.17648e7	0.447244	0.223622	−	0.974676i	$-0.428212\pi$
$929$	1.17648e7	0.447244	0.223622	+	0.974676i	$0.428212\pi$
$930$	0	0
$931$	0	0
$932$	−2.10792e7	−0.794905
$933$	0	0
$934$	1.65835e7	0.622026
$935$	−6.58886e7	−2.46480
$936$	0	0
$937$	7.83051e6	0.291367	0.145684	−	0.989331i	$-0.453462\pi$
$937$	7.83051e6	0.291367	0.145684	+	0.989331i	$0.453462\pi$
$938$	0	0
$939$	0	0
$940$	−1.97711e7	−0.729812
$941$	−3.39315e7	−1.24919	−0.624595	−	0.780948i	$-0.714736\pi$
$941$	−3.39315e7	−1.24919	−0.624595	+	0.780948i	$0.714736\pi$
$942$	0	0
$943$	7.47849e7	2.73864
$944$	−4.30718e6	−0.157312
$945$	0	0
$946$	4.03974e6	0.146766
$947$	4.08599e7	1.48055	0.740274	−	0.672305i	$-0.234696\pi$
$947$	4.08599e7	1.48055	0.740274	+	0.672305i	$0.234696\pi$
$948$	0	0
$949$	−3.21136e7	−1.15751
$950$	−2.18267e6	−0.0784655
$951$	0	0
$952$	0	0
$953$	3.53837e7	1.26203	0.631016	−	0.775770i	$-0.282638\pi$
$953$	3.53837e7	1.26203	0.631016	+	0.775770i	$0.282638\pi$
$954$	0	0
$955$	3.50296e7	1.24287
$956$	−1.28107e7	−0.453345
$957$	0	0
$958$	−1.03071e6	−0.0362846
$959$	0	0
$960$	0	0
$961$	−2.82161e7	−0.985571
$962$	3.94947e6	0.137594
$963$	0	0
$964$	6.43691e6	0.223093
$965$	−2.18623e7	−0.755750
$966$	0	0
$967$	−4.37744e7	−1.50541	−0.752703	−	0.658360i	$-0.771250\pi$
$967$	−4.37744e7	−1.50541	−0.752703	+	0.658360i	$0.771250\pi$
$968$	−1.40045e7	−0.480374
$969$	0	0
$970$	−5.80679e6	−0.198156
$971$	7.16196e6	0.243772	0.121886	−	0.992544i	$-0.461106\pi$
$971$	7.16196e6	0.243772	0.121886	+	0.992544i	$0.461106\pi$
$972$	0	0
$973$	0	0
$974$	−6.88132e6	−0.232421
$975$	0	0
$976$	−3.46296e6	−0.116365
$977$	2.68803e7	0.900945	0.450473	−	0.892790i	$-0.351255\pi$
$977$	2.68803e7	0.900945	0.450473	+	0.892790i	$0.351255\pi$
$978$	0	0
$979$	−4.57078e7	−1.52417
$980$	0	0
$981$	0	0
$982$	2.59641e7	0.859199
$983$	−1.06207e6	−0.0350565	−0.0175283	−	0.999846i	$-0.505580\pi$
$983$	−1.06207e6	−0.0350565	−0.0175283	+	0.999846i	$0.505580\pi$
$984$	0	0
$985$	−4.33522e7	−1.42371
$986$	−1.23135e7	−0.403355
$987$	0	0
$988$	8.56779e6	0.279239
$989$	7.83531e6	0.254721
$990$	0	0
$991$	−7.39380e6	−0.239157	−0.119579	−	0.992825i	$-0.538154\pi$
$991$	−7.39380e6	−0.239157	−0.119579	+	0.992825i	$0.538154\pi$
$992$	−658147.	−0.0212346
$993$	0	0
$994$	0	0
$995$	−1.39931e7	−0.448082
$996$	0	0
$997$	2.41365e7	0.769019	0.384510	−	0.923121i	$-0.374371\pi$
$997$	2.41365e7	0.769019	0.384510	+	0.923121i	$0.374371\pi$
$998$	−1.51852e6	−0.0482609
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.bc.1.2		2
3.2	odd	2		294.6.a.v.1.1	yes	2
7.6	odd	2		882.6.a.bg.1.1		2
21.2	odd	6		294.6.e.t.67.2		4
21.5	even	6		294.6.e.w.67.1		4
21.11	odd	6		294.6.e.t.79.2		4
21.17	even	6		294.6.e.w.79.1		4
21.20	even	2		294.6.a.s.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.6.a.s.1.2	✓	2	21.20	even	2
294.6.a.v.1.1	yes	2	3.2	odd	2
294.6.e.t.67.2		4	21.2	odd	6
294.6.e.t.79.2		4	21.11	odd	6
294.6.e.w.67.1		4	21.5	even	6
294.6.e.w.79.1		4	21.17	even	6
882.6.a.bc.1.2		2	1.1	even	1	trivial
882.6.a.bg.1.1		2	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +59.5930 q^{5} -64.0000 q^{8} +O(q^{10})\)	\(q-4.00000 q^{2} +16.0000 q^{4} +59.5930 q^{5} -64.0000 q^{8} -238.372 q^{10} +616.337 q^{11} +418.372 q^{13} +256.000 q^{16} -1793.90 q^{17} +1279.93 q^{19} +953.488 q^{20} -2465.35 q^{22} -4781.69 q^{23} +426.326 q^{25} -1673.49 q^{26} -1716.02 q^{29} +642.722 q^{31} -1024.00 q^{32} +7175.58 q^{34} -2360.02 q^{37} -5119.72 q^{38} -3813.95 q^{40} -15639.9 q^{41} -1638.61 q^{43} +9861.39 q^{44} +19126.7 q^{46} -20735.5 q^{47} -1705.30 q^{50} +6693.95 q^{52} +5347.93 q^{53} +36729.4 q^{55} +6864.09 q^{58} -16824.9 q^{59} -13527.2 q^{61} -2570.89 q^{62} +4096.00 q^{64} +24932.0 q^{65} -46483.5 q^{67} -28702.3 q^{68} +3273.91 q^{71} -76758.4 q^{73} +9440.09 q^{74} +20478.9 q^{76} -17731.4 q^{79} +15255.8 q^{80} +62559.5 q^{82} -78846.7 q^{83} -106904. q^{85} +6554.43 q^{86} -39445.6 q^{88} -74160.4 q^{89} -76507.0 q^{92} +82942.1 q^{94} +76274.9 q^{95} +24360.2 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} - 18 q^{5} - 128 q^{8}+O(q^{10}) \) 2 * q - 8 * q^2 + 32 * q^4 - 18 * q^5 - 128 * q^8	\( 2 q - 8 q^{2} + 32 q^{4} - 18 q^{5} - 128 q^{8} + 72 q^{10} - 2 q^{11} + 288 q^{13} + 512 q^{16} - 1530 q^{17} + 1188 q^{19} - 288 q^{20} + 8 q^{22} - 3390 q^{23} + 3322 q^{25} - 1152 q^{26} + 3976 q^{29} + 7596 q^{31} - 2048 q^{32} + 6120 q^{34} + 2688 q^{37} - 4752 q^{38} + 1152 q^{40} - 36630 q^{41} - 23032 q^{43} - 32 q^{44} + 13560 q^{46} - 864 q^{47} - 13288 q^{50} + 4608 q^{52} + 32920 q^{53} + 84708 q^{55} - 15904 q^{58} + 26712 q^{59} + 20412 q^{61} - 30384 q^{62} + 8192 q^{64} + 35048 q^{65} - 36172 q^{67} - 24480 q^{68} - 73706 q^{71} - 74772 q^{73} - 10752 q^{74} + 19008 q^{76} - 23116 q^{79} - 4608 q^{80} + 146520 q^{82} - 147816 q^{83} - 127380 q^{85} + 92128 q^{86} + 128 q^{88} - 164646 q^{89} - 54240 q^{92} + 3456 q^{94} + 83408 q^{95} + 162036 q^{97}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 - 18 * q^5 - 128 * q^8 + 72 * q^10 - 2 * q^11 + 288 * q^13 + 512 * q^16 - 1530 * q^17 + 1188 * q^19 - 288 * q^20 + 8 * q^22 - 3390 * q^23 + 3322 * q^25 - 1152 * q^26 + 3976 * q^29 + 7596 * q^31 - 2048 * q^32 + 6120 * q^34 + 2688 * q^37 - 4752 * q^38 + 1152 * q^40 - 36630 * q^41 - 23032 * q^43 - 32 * q^44 + 13560 * q^46 - 864 * q^47 - 13288 * q^50 + 4608 * q^52 + 32920 * q^53 + 84708 * q^55 - 15904 * q^58 + 26712 * q^59 + 20412 * q^61 - 30384 * q^62 + 8192 * q^64 + 35048 * q^65 - 36172 * q^67 - 24480 * q^68 - 73706 * q^71 - 74772 * q^73 - 10752 * q^74 + 19008 * q^76 - 23116 * q^79 - 4608 * q^80 + 146520 * q^82 - 147816 * q^83 - 127380 * q^85 + 92128 * q^86 + 128 * q^88 - 164646 * q^89 - 54240 * q^92 + 3456 * q^94 + 83408 * q^95 + 162036 * q^97

Introduction

L-functions

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