LMFDB - Embedded newform 475.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,6,Mod(1,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, 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("475.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	475.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:	$76.1823144112$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

\(q+7.00000 q^{2} +11.0000 q^{3} +17.0000 q^{4} +77.0000 q^{6} +197.000 q^{7} -105.000 q^{8} -122.000 q^{9} -468.000 q^{11} +187.000 q^{12} +921.000 q^{13} +1379.00 q^{14} -1279.00 q^{16} +1107.00 q^{17} -854.000 q^{18} +361.000 q^{19} +2167.00 q^{21} -3276.00 q^{22} +3641.00 q^{23} -1155.00 q^{24} +6447.00 q^{26} -4015.00 q^{27} +3349.00 q^{28} +7525.00 q^{29} +1422.00 q^{31} -5593.00 q^{32} -5148.00 q^{33} +7749.00 q^{34} -2074.00 q^{36} +11282.0 q^{37} +2527.00 q^{38} +10131.0 q^{39} -678.000 q^{41} +15169.0 q^{42} -5974.00 q^{43} -7956.00 q^{44} +25487.0 q^{46} +11072.0 q^{47} -14069.0 q^{48} +22002.0 q^{49} +12177.0 q^{51} +15657.0 q^{52} +17461.0 q^{53} -28105.0 q^{54} -20685.0 q^{56} +3971.00 q^{57} +52675.0 q^{58} -46305.0 q^{59} +16292.0 q^{61} +9954.00 q^{62} -24034.0 q^{63} +1777.00 q^{64} -36036.0 q^{66} -36373.0 q^{67} +18819.0 q^{68} +40051.0 q^{69} -82208.0 q^{71} +12810.0 q^{72} +43861.0 q^{73} +78974.0 q^{74} +6137.00 q^{76} -92196.0 q^{77} +70917.0 q^{78} -30130.0 q^{79} -14519.0 q^{81} -4746.00 q^{82} +91626.0 q^{83} +36839.0 q^{84} -41818.0 q^{86} +82775.0 q^{87} +49140.0 q^{88} +79170.0 q^{89} +181437. q^{91} +61897.0 q^{92} +15642.0 q^{93} +77504.0 q^{94} -61523.0 q^{96} -128718. q^{97} +154014. q^{98} +57096.0 q^{99} +O(q^{100})\)

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 $

$2$	7.00000	1.23744	0.618718	−	0.785613i	$-0.287652\pi$
$2$	7.00000	1.23744	0.618718	+	0.785613i	$0.287652\pi$
$3$	11.0000	0.705650	0.352825	−	0.935689i	$-0.385221\pi$
$3$	11.0000	0.705650	0.352825	+	0.935689i	$0.385221\pi$
$4$	17.0000	0.531250
$5$	0	0
$5$	0	0
$6$	77.0000	0.873198
$7$	197.000	1.51957	0.759786	−	0.650174i	$-0.225304\pi$
$7$	197.000	1.51957	0.759786	+	0.650174i	$0.225304\pi$
$8$	−105.000	−0.580049
$9$	−122.000	−0.502058
$10$	0	0
$11$	−468.000	−1.16618	−0.583088	−	0.812409i	$-0.698156\pi$
$11$	−468.000	−1.16618	−0.583088	+	0.812409i	$0.698156\pi$
$12$	187.000	0.374877
$13$	921.000	1.51148	0.755738	−	0.654874i	$-0.227278\pi$
$13$	921.000	1.51148	0.755738	+	0.654874i	$0.227278\pi$
$14$	1379.00	1.88037
$15$	0	0
$16$	−1279.00	−1.24902
$17$	1107.00	0.929021	0.464510	−	0.885568i	$-0.346230\pi$
$17$	1107.00	0.929021	0.464510	+	0.885568i	$0.346230\pi$
$18$	−854.000	−0.621265
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	2167.00	1.07229
$22$	−3276.00	−1.44307
$23$	3641.00	1.43516	0.717581	−	0.696475i	$-0.245249\pi$
$23$	3641.00	1.43516	0.717581	+	0.696475i	$0.245249\pi$
$24$	−1155.00	−0.409311
$25$	0	0
$26$	6447.00	1.87036
$27$	−4015.00	−1.05993
$28$	3349.00	0.807272
$29$	7525.00	1.66154	0.830771	−	0.556614i	$-0.187900\pi$
$29$	7525.00	1.66154	0.830771	+	0.556614i	$0.187900\pi$
$30$	0	0
$31$	1422.00	0.265764	0.132882	−	0.991132i	$-0.457577\pi$
$31$	1422.00	0.265764	0.132882	+	0.991132i	$0.457577\pi$
$32$	−5593.00	−0.965539
$33$	−5148.00	−0.822913
$34$	7749.00	1.14960
$35$	0	0
$36$	−2074.00	−0.266718
$37$	11282.0	1.35482	0.677410	−	0.735605i	$-0.263102\pi$
$37$	11282.0	1.35482	0.677410	+	0.735605i	$0.263102\pi$
$38$	2527.00	0.283887
$39$	10131.0	1.06657
$40$	0	0
$41$	−678.000	−0.0629898	−0.0314949	−	0.999504i	$-0.510027\pi$
$41$	−678.000	−0.0629898	−0.0314949	+	0.999504i	$0.510027\pi$
$42$	15169.0	1.32689
$43$	−5974.00	−0.492713	−0.246357	−	0.969179i	$-0.579233\pi$
$43$	−5974.00	−0.492713	−0.246357	+	0.969179i	$0.579233\pi$
$44$	−7956.00	−0.619531
$45$	0	0
$46$	25487.0	1.77592
$47$	11072.0	0.731108	0.365554	−	0.930790i	$-0.380880\pi$
$47$	11072.0	0.731108	0.365554	+	0.930790i	$0.380880\pi$
$48$	−14069.0	−0.881374
$49$	22002.0	1.30910
$50$	0	0
$51$	12177.0	0.655564
$52$	15657.0	0.802972
$53$	17461.0	0.853846	0.426923	−	0.904288i	$-0.359598\pi$
$53$	17461.0	0.853846	0.426923	+	0.904288i	$0.359598\pi$
$54$	−28105.0	−1.31159
$55$	0	0
$56$	−20685.0	−0.881425
$57$	3971.00	0.161887
$58$	52675.0	2.05605
$59$	−46305.0	−1.73180	−0.865900	−	0.500217i	$-0.833254\pi$
$59$	−46305.0	−1.73180	−0.865900	+	0.500217i	$0.833254\pi$
$60$	0	0
$61$	16292.0	0.560596	0.280298	−	0.959913i	$-0.409567\pi$
$61$	16292.0	0.560596	0.280298	+	0.959913i	$0.409567\pi$
$62$	9954.00	0.328866
$63$	−24034.0	−0.762912
$64$	1777.00	0.0542297
$65$	0	0
$66$	−36036.0	−1.01830
$67$	−36373.0	−0.989902	−0.494951	−	0.868921i	$-0.664814\pi$
$67$	−36373.0	−0.989902	−0.494951	+	0.868921i	$0.664814\pi$
$68$	18819.0	0.493542
$69$	40051.0	1.01272
$70$	0	0
$71$	−82208.0	−1.93539	−0.967694	−	0.252126i	$-0.918870\pi$
$71$	−82208.0	−1.93539	−0.967694	+	0.252126i	$0.918870\pi$
$72$	12810.0	0.291218
$73$	43861.0	0.963322	0.481661	−	0.876358i	$-0.340034\pi$
$73$	43861.0	0.963322	0.481661	+	0.876358i	$0.340034\pi$
$74$	78974.0	1.67650
$75$	0	0
$76$	6137.00	0.121877
$77$	−92196.0	−1.77209
$78$	70917.0	1.31982
$79$	−30130.0	−0.543165	−0.271582	−	0.962415i	$-0.587547\pi$
$79$	−30130.0	−0.543165	−0.271582	+	0.962415i	$0.587547\pi$
$80$	0	0
$81$	−14519.0	−0.245881
$82$	−4746.00	−0.0779459
$83$	91626.0	1.45990	0.729951	−	0.683500i	$-0.239543\pi$
$83$	91626.0	1.45990	0.729951	+	0.683500i	$0.239543\pi$
$84$	36839.0	0.569652
$85$	0	0
$86$	−41818.0	−0.609701
$87$	82775.0	1.17247
$88$	49140.0	0.676439
$89$	79170.0	1.05946	0.529731	−	0.848166i	$-0.322293\pi$
$89$	79170.0	1.05946	0.529731	+	0.848166i	$0.322293\pi$
$90$	0	0
$91$	181437.	2.29680
$92$	61897.0	0.762430
$93$	15642.0	0.187536
$94$	77504.0	0.904700
$95$	0	0
$96$	−61523.0	−0.681333
$97$	−128718.	−1.38902	−0.694512	−	0.719481i	$-0.744380\pi$
$97$	−128718.	−1.38902	−0.694512	+	0.719481i	$0.744380\pi$
$98$	154014.	1.61993
$99$	57096.0	0.585488
$100$	0	0
$101$	63432.0	0.618735	0.309368	−	0.950942i	$-0.399883\pi$
$101$	63432.0	0.618735	0.309368	+	0.950942i	$0.399883\pi$
$102$	85239.0	0.811219
$103$	49436.0	0.459145	0.229573	−	0.973292i	$-0.426267\pi$
$103$	49436.0	0.459145	0.229573	+	0.973292i	$0.426267\pi$
$104$	−96705.0	−0.876729
$105$	0	0
$106$	122227.	1.05658
$107$	−42603.0	−0.359733	−0.179867	−	0.983691i	$-0.557567\pi$
$107$	−42603.0	−0.359733	−0.179867	+	0.983691i	$0.557567\pi$
$108$	−68255.0	−0.563086
$109$	41355.0	0.333397	0.166699	−	0.986008i	$-0.446689\pi$
$109$	41355.0	0.333397	0.166699	+	0.986008i	$0.446689\pi$
$110$	0	0
$111$	124102.	0.956030
$112$	−251963.	−1.89798
$113$	172896.	1.27376	0.636882	−	0.770961i	$-0.280224\pi$
$113$	172896.	1.27376	0.636882	+	0.770961i	$0.280224\pi$
$114$	27797.0	0.200325
$115$	0	0
$116$	127925.	0.882695
$117$	−112362.	−0.758848
$118$	−324135.	−2.14299
$119$	218079.	1.41171
$120$	0	0
$121$	57973.0	0.359967
$122$	114044.	0.693702
$123$	−7458.00	−0.0444488
$124$	24174.0	0.141187
$125$	0	0
$126$	−168238.	−0.944056
$127$	−190138.	−1.04607	−0.523034	−	0.852312i	$-0.675200\pi$
$127$	−190138.	−1.04607	−0.523034	+	0.852312i	$0.675200\pi$
$128$	191415.	1.03264
$129$	−65714.0	−0.347683
$130$	0	0
$131$	−184448.	−0.939065	−0.469533	−	0.882915i	$-0.655578\pi$
$131$	−184448.	−0.939065	−0.469533	+	0.882915i	$0.655578\pi$
$132$	−87516.0	−0.437172
$133$	71117.0	0.348614
$134$	−254611.	−1.22494
$135$	0	0
$136$	−116235.	−0.538877
$137$	274807.	1.25091	0.625455	−	0.780260i	$-0.284913\pi$
$137$	274807.	1.25091	0.625455	+	0.780260i	$0.284913\pi$
$138$	280357.	1.25318
$139$	−337360.	−1.48101	−0.740503	−	0.672053i	$-0.765412\pi$
$139$	−337360.	−1.48101	−0.740503	+	0.672053i	$0.765412\pi$
$140$	0	0
$141$	121792.	0.515906
$142$	−575456.	−2.39492
$143$	−431028.	−1.76265
$144$	156038.	0.627082
$145$	0	0
$146$	307027.	1.19205
$147$	242022.	0.923765
$148$	191794.	0.719748
$149$	−54420.0	−0.200813	−0.100407	−	0.994946i	$-0.532014\pi$
$149$	−54420.0	−0.200813	−0.100407	+	0.994946i	$0.532014\pi$
$150$	0	0
$151$	−487498.	−1.73992	−0.869962	−	0.493118i	$-0.835857\pi$
$151$	−487498.	−1.73992	−0.869962	+	0.493118i	$0.835857\pi$
$152$	−37905.0	−0.133072
$153$	−135054.	−0.466422
$154$	−645372.	−2.19285
$155$	0	0
$156$	172227.	0.566617
$157$	339482.	1.09918	0.549588	−	0.835436i	$-0.314785\pi$
$157$	339482.	1.09918	0.549588	+	0.835436i	$0.314785\pi$
$158$	−210910.	−0.672132
$159$	192071.	0.602517
$160$	0	0
$161$	717277.	2.18083
$162$	−101633.	−0.304262
$163$	−300374.	−0.885510	−0.442755	−	0.896643i	$-0.645999\pi$
$163$	−300374.	−0.885510	−0.442755	+	0.896643i	$0.645999\pi$
$164$	−11526.0	−0.0334633
$165$	0	0
$166$	641382.	1.80654
$167$	165122.	0.458156	0.229078	−	0.973408i	$-0.426429\pi$
$167$	165122.	0.458156	0.229078	+	0.973408i	$0.426429\pi$
$168$	−227535.	−0.621978
$169$	476948.	1.28456
$170$	0	0
$171$	−44042.0	−0.115180
$172$	−101558.	−0.261754
$173$	581406.	1.47694	0.738472	−	0.674284i	$-0.235548\pi$
$173$	581406.	1.47694	0.738472	+	0.674284i	$0.235548\pi$
$174$	579425.	1.45086
$175$	0	0
$176$	598572.	1.45658
$177$	−509355.	−1.22205
$178$	554190.	1.31102
$179$	150520.	0.351125	0.175562	−	0.984468i	$-0.443826\pi$
$179$	150520.	0.351125	0.175562	+	0.984468i	$0.443826\pi$
$180$	0	0
$181$	−208598.	−0.473275	−0.236638	−	0.971598i	$-0.576045\pi$
$181$	−208598.	−0.473275	−0.236638	+	0.971598i	$0.576045\pi$
$182$	1.27006e6	2.84214
$183$	179212.	0.395585
$184$	−382305.	−0.832464
$185$	0	0
$186$	109494.	0.232064
$187$	−518076.	−1.08340
$188$	188224.	0.388401
$189$	−790955.	−1.61064
$190$	0	0
$191$	−853463.	−1.69278	−0.846391	−	0.532561i	$-0.821230\pi$
$191$	−853463.	−1.69278	−0.846391	+	0.532561i	$0.821230\pi$
$192$	19547.0	0.0382672
$193$	−229674.	−0.443832	−0.221916	−	0.975066i	$-0.571231\pi$
$193$	−229674.	−0.443832	−0.221916	+	0.975066i	$0.571231\pi$
$194$	−901026.	−1.71883
$195$	0	0
$196$	374034.	0.695458
$197$	−842778.	−1.54720	−0.773602	−	0.633672i	$-0.781547\pi$
$197$	−842778.	−1.54720	−0.773602	+	0.633672i	$0.781547\pi$
$198$	399672.	0.724504
$199$	636865.	1.14003	0.570013	−	0.821636i	$-0.306938\pi$
$199$	636865.	1.14003	0.570013	+	0.821636i	$0.306938\pi$
$200$	0	0
$201$	−400103.	−0.698525
$202$	444024.	0.765646
$203$	1.48242e6	2.52483
$204$	207009.	0.348268
$205$	0	0
$206$	346052.	0.568163
$207$	−444202.	−0.720534
$208$	−1.17796e6	−1.88787
$209$	−168948.	−0.267539
$210$	0	0
$211$	−684173.	−1.05794	−0.528968	−	0.848641i	$-0.677421\pi$
$211$	−684173.	−1.05794	−0.528968	+	0.848641i	$0.677421\pi$
$212$	296837.	0.453606
$213$	−904288.	−1.36571
$214$	−298221.	−0.445147
$215$	0	0
$216$	421575.	0.614809
$217$	280134.	0.403847
$218$	289485.	0.412558
$219$	482471.	0.679768
$220$	0	0
$221$	1.01955e6	1.40419
$222$	868714.	1.18303
$223$	−370094.	−0.498368	−0.249184	−	0.968456i	$-0.580162\pi$
$223$	−370094.	−0.498368	−0.249184	+	0.968456i	$0.580162\pi$
$224$	−1.10182e6	−1.46721
$225$	0	0
$226$	1.21027e6	1.57620
$227$	497827.	0.641230	0.320615	−	0.947210i	$-0.396110\pi$
$227$	497827.	0.641230	0.320615	+	0.947210i	$0.396110\pi$
$228$	67507.0	0.0860026
$229$	−848470.	−1.06917	−0.534586	−	0.845114i	$-0.679533\pi$
$229$	−848470.	−1.06917	−0.534586	+	0.845114i	$0.679533\pi$
$230$	0	0
$231$	−1.01416e6	−1.25047
$232$	−790125.	−0.963775
$233$	153206.	0.184878	0.0924392	−	0.995718i	$-0.470534\pi$
$233$	153206.	0.184878	0.0924392	+	0.995718i	$0.470534\pi$
$234$	−786534.	−0.939027
$235$	0	0
$236$	−787185.	−0.920019
$237$	−331430.	−0.383284
$238$	1.52655e6	1.74691
$239$	−325035.	−0.368074	−0.184037	−	0.982919i	$-0.558917\pi$
$239$	−325035.	−0.368074	−0.184037	+	0.982919i	$0.558917\pi$
$240$	0	0
$241$	870852.	0.965832	0.482916	−	0.875667i	$-0.339578\pi$
$241$	870852.	0.965832	0.482916	+	0.875667i	$0.339578\pi$
$242$	405811.	0.445436
$243$	815936.	0.886422
$244$	276964.	0.297817
$245$	0	0
$246$	−52206.0	−0.0550025
$247$	332481.	0.346756
$248$	−149310.	−0.154156
$249$	1.00789e6	1.03018
$250$	0	0
$251$	973812.	0.975643	0.487821	−	0.872943i	$-0.337792\pi$
$251$	973812.	0.975643	0.487821	+	0.872943i	$0.337792\pi$
$252$	−408578.	−0.405297
$253$	−1.70399e6	−1.67365
$254$	−1.33097e6	−1.29444
$255$	0	0
$256$	1.28304e6	1.22360
$257$	−1.38172e6	−1.30493	−0.652464	−	0.757820i	$-0.726265\pi$
$257$	−1.38172e6	−1.30493	−0.652464	+	0.757820i	$0.726265\pi$
$258$	−459998.	−0.430236
$259$	2.22255e6	2.05875
$260$	0	0
$261$	−918050.	−0.834190
$262$	−1.29114e6	−1.16203
$263$	−449064.	−0.400331	−0.200165	−	0.979762i	$-0.564148\pi$
$263$	−449064.	−0.400331	−0.200165	+	0.979762i	$0.564148\pi$
$264$	540540.	0.477329
$265$	0	0
$266$	497819.	0.431387
$267$	870870.	0.747610
$268$	−618341.	−0.525885
$269$	59250.0	0.0499238	0.0249619	−	0.999688i	$-0.492054\pi$
$269$	59250.0	0.0499238	0.0249619	+	0.999688i	$0.492054\pi$
$270$	0	0
$271$	−194843.	−0.161162	−0.0805808	−	0.996748i	$-0.525678\pi$
$271$	−194843.	−0.161162	−0.0805808	+	0.996748i	$0.525678\pi$
$272$	−1.41585e6	−1.16037
$273$	1.99581e6	1.62073
$274$	1.92365e6	1.54792
$275$	0	0
$276$	680867.	0.538009
$277$	−65848.0	−0.0515636	−0.0257818	−	0.999668i	$-0.508208\pi$
$277$	−65848.0	−0.0515636	−0.0257818	+	0.999668i	$0.508208\pi$
$278$	−2.36152e6	−1.83265
$279$	−173484.	−0.133429
$280$	0	0
$281$	270302.	0.204213	0.102107	−	0.994773i	$-0.467442\pi$
$281$	270302.	0.204213	0.102107	+	0.994773i	$0.467442\pi$
$282$	852544.	0.638402
$283$	−802174.	−0.595391	−0.297696	−	0.954661i	$-0.596218\pi$
$283$	−802174.	−0.595391	−0.297696	+	0.954661i	$0.596218\pi$
$284$	−1.39754e6	−1.02818
$285$	0	0
$286$	−3.01720e6	−2.18116
$287$	−133566.	−0.0957175
$288$	682346.	0.484756
$289$	−194408.	−0.136921
$290$	0	0
$291$	−1.41590e6	−0.980166
$292$	745637.	0.511765
$293$	−533229.	−0.362865	−0.181432	−	0.983403i	$-0.558073\pi$
$293$	−533229.	−0.362865	−0.181432	+	0.983403i	$0.558073\pi$
$294$	1.69415e6	1.14310
$295$	0	0
$296$	−1.18461e6	−0.785862
$297$	1.87902e6	1.23606
$298$	−380940.	−0.248494
$299$	3.35336e6	2.16921
$300$	0	0
$301$	−1.17688e6	−0.748713
$302$	−3.41249e6	−2.15305
$303$	697752.	0.436611
$304$	−461719.	−0.286546
$305$	0	0
$306$	−945378.	−0.577168
$307$	1.19721e6	0.724978	0.362489	−	0.931988i	$-0.381927\pi$
$307$	1.19721e6	0.724978	0.362489	+	0.931988i	$0.381927\pi$
$308$	−1.56733e6	−0.941422
$309$	543796.	0.323996
$310$	0	0
$311$	−1.42130e6	−0.833270	−0.416635	−	0.909074i	$-0.636791\pi$
$311$	−1.42130e6	−0.833270	−0.416635	+	0.909074i	$0.636791\pi$
$312$	−1.06375e6	−0.618664
$313$	−1.39904e6	−0.807177	−0.403589	−	0.914941i	$-0.632237\pi$
$313$	−1.39904e6	−0.807177	−0.403589	+	0.914941i	$0.632237\pi$
$314$	2.37637e6	1.36016
$315$	0	0
$316$	−512210.	−0.288556
$317$	−780993.	−0.436515	−0.218257	−	0.975891i	$-0.570037\pi$
$317$	−780993.	−0.436515	−0.218257	+	0.975891i	$0.570037\pi$
$318$	1.34450e6	0.745576
$319$	−3.52170e6	−1.93765
$320$	0	0
$321$	−468633.	−0.253846
$322$	5.02094e6	2.69864
$323$	399627.	0.213132
$324$	−246823.	−0.130624
$325$	0	0
$326$	−2.10262e6	−1.09576
$327$	454905.	0.235262
$328$	71190.0	0.0365371
$329$	2.18118e6	1.11097
$330$	0	0
$331$	−2.95838e6	−1.48417	−0.742086	−	0.670304i	$-0.766164\pi$
$331$	−2.95838e6	−1.48417	−0.742086	+	0.670304i	$0.766164\pi$
$332$	1.55764e6	0.775573
$333$	−1.37640e6	−0.680198
$334$	1.15585e6	0.566940
$335$	0	0
$336$	−2.77159e6	−1.33931
$337$	−2.16116e6	−1.03660	−0.518301	−	0.855198i	$-0.673435\pi$
$337$	−2.16116e6	−1.03660	−0.518301	+	0.855198i	$0.673435\pi$
$338$	3.33864e6	1.58956
$339$	1.90186e6	0.898832
$340$	0	0
$341$	−665496.	−0.309927
$342$	−308294.	−0.142528
$343$	1.02341e6	0.469696
$344$	627270.	0.285797
$345$	0	0
$346$	4.06984e6	1.82763
$347$	1.83170e6	0.816641	0.408320	−	0.912839i	$-0.366115\pi$
$347$	1.83170e6	0.816641	0.408320	+	0.912839i	$0.366115\pi$
$348$	1.40718e6	0.622874
$349$	1.83019e6	0.804327	0.402163	−	0.915568i	$-0.368258\pi$
$349$	1.83019e6	0.804327	0.402163	+	0.915568i	$0.368258\pi$
$350$	0	0
$351$	−3.69782e6	−1.60205
$352$	2.61752e6	1.12599
$353$	1.01076e6	0.431729	0.215865	−	0.976423i	$-0.430743\pi$
$353$	1.01076e6	0.431729	0.215865	+	0.976423i	$0.430743\pi$
$354$	−3.56548e6	−1.51220
$355$	0	0
$356$	1.34589e6	0.562840
$357$	2.39887e6	0.996176
$358$	1.05364e6	0.434495
$359$	−1.97064e6	−0.806998	−0.403499	−	0.914980i	$-0.632206\pi$
$359$	−1.97064e6	−0.806998	−0.403499	+	0.914980i	$0.632206\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	−1.46019e6	−0.585648
$363$	637703.	0.254011
$364$	3.08443e6	1.22017
$365$	0	0
$366$	1.25448e6	0.489511
$367$	3.63567e6	1.40903	0.704514	−	0.709690i	$-0.251165\pi$
$367$	3.63567e6	1.40903	0.704514	+	0.709690i	$0.251165\pi$
$368$	−4.65684e6	−1.79255
$369$	82716.0	0.0316245
$370$	0	0
$371$	3.43982e6	1.29748
$372$	265914.	0.0996286
$373$	466191.	0.173497	0.0867485	−	0.996230i	$-0.472352\pi$
$373$	466191.	0.173497	0.0867485	+	0.996230i	$0.472352\pi$
$374$	−3.62653e6	−1.34064
$375$	0	0
$376$	−1.16256e6	−0.424078
$377$	6.93053e6	2.51138
$378$	−5.53668e6	−1.99306
$379$	−1.88786e6	−0.675107	−0.337554	−	0.941306i	$-0.609599\pi$
$379$	−1.88786e6	−0.675107	−0.337554	+	0.941306i	$0.609599\pi$
$380$	0	0
$381$	−2.09152e6	−0.738158
$382$	−5.97424e6	−2.09471
$383$	4.53236e6	1.57880	0.789400	−	0.613879i	$-0.210392\pi$
$383$	4.53236e6	1.57880	0.789400	+	0.613879i	$0.210392\pi$
$384$	2.10556e6	0.728686
$385$	0	0
$386$	−1.60772e6	−0.549214
$387$	728828.	0.247370
$388$	−2.18821e6	−0.737919
$389$	−2.49266e6	−0.835197	−0.417599	−	0.908632i	$-0.637128\pi$
$389$	−2.49266e6	−0.835197	−0.417599	+	0.908632i	$0.637128\pi$
$390$	0	0
$391$	4.03059e6	1.33330
$392$	−2.31021e6	−0.759340
$393$	−2.02893e6	−0.662652
$394$	−5.89945e6	−1.91457
$395$	0	0
$396$	970632.	0.311040
$397$	−4.71605e6	−1.50176	−0.750882	−	0.660436i	$-0.770372\pi$
$397$	−4.71605e6	−1.50176	−0.750882	+	0.660436i	$0.770372\pi$
$398$	4.45806e6	1.41071
$399$	782287.	0.245999
$400$	0	0
$401$	−3.48844e6	−1.08335	−0.541677	−	0.840587i	$-0.682210\pi$
$401$	−3.48844e6	−1.08335	−0.541677	+	0.840587i	$0.682210\pi$
$402$	−2.80072e6	−0.864380
$403$	1.30966e6	0.401695
$404$	1.07834e6	0.328703
$405$	0	0
$406$	1.03770e7	3.12432
$407$	−5.27998e6	−1.57996
$408$	−1.27859e6	−0.380259
$409$	−2.27305e6	−0.671894	−0.335947	−	0.941881i	$-0.609056\pi$
$409$	−2.27305e6	−0.671894	−0.335947	+	0.941881i	$0.609056\pi$
$410$	0	0
$411$	3.02288e6	0.882706
$412$	840412.	0.243921
$413$	−9.12208e6	−2.63159
$414$	−3.10941e6	−0.891616
$415$	0	0
$416$	−5.15115e6	−1.45939
$417$	−3.71096e6	−1.04507
$418$	−1.18264e6	−0.331063
$419$	4.18803e6	1.16540	0.582700	−	0.812688i	$-0.301996\pi$
$419$	4.18803e6	1.16540	0.582700	+	0.812688i	$0.301996\pi$
$420$	0	0
$421$	−1.10202e6	−0.303030	−0.151515	−	0.988455i	$-0.548415\pi$
$421$	−1.10202e6	−0.303030	−0.151515	+	0.988455i	$0.548415\pi$
$422$	−4.78921e6	−1.30913
$423$	−1.35078e6	−0.367058
$424$	−1.83340e6	−0.495272
$425$	0	0
$426$	−6.33002e6	−1.68998
$427$	3.20952e6	0.851865
$428$	−724251.	−0.191108
$429$	−4.74131e6	−1.24381
$430$	0	0
$431$	69612.0	0.0180506	0.00902529	−	0.999959i	$-0.497127\pi$
$431$	69612.0	0.0180506	0.00902529	+	0.999959i	$0.497127\pi$
$432$	5.13518e6	1.32387
$433$	−4.49422e6	−1.15195	−0.575977	−	0.817466i	$-0.695378\pi$
$433$	−4.49422e6	−1.15195	−0.575977	+	0.817466i	$0.695378\pi$
$434$	1.96094e6	0.499735
$435$	0	0
$436$	703035.	0.177117
$437$	1.31440e6	0.329249
$438$	3.37730e6	0.841170
$439$	−161910.	−0.0400970	−0.0200485	−	0.999799i	$-0.506382\pi$
$439$	−161910.	−0.0400970	−0.0200485	+	0.999799i	$0.506382\pi$
$440$	0	0
$441$	−2.68424e6	−0.657242
$442$	7.13683e6	1.73760
$443$	−8.03624e6	−1.94556	−0.972778	−	0.231738i	$-0.925559\pi$
$443$	−8.03624e6	−1.94556	−0.972778	+	0.231738i	$0.925559\pi$
$444$	2.10973e6	0.507891
$445$	0	0
$446$	−2.59066e6	−0.616699
$447$	−598620.	−0.141704
$448$	350069.	0.0824060
$449$	8.19457e6	1.91827	0.959136	−	0.282944i	$-0.0913112\pi$
$449$	8.19457e6	1.91827	0.959136	+	0.282944i	$0.0913112\pi$
$450$	0	0
$451$	317304.	0.0734572
$452$	2.93923e6	0.676687
$453$	−5.36248e6	−1.22778
$454$	3.48479e6	0.793482
$455$	0	0
$456$	−416955.	−0.0939025
$457$	−1.39632e6	−0.312749	−0.156374	−	0.987698i	$-0.549981\pi$
$457$	−1.39632e6	−0.312749	−0.156374	+	0.987698i	$0.549981\pi$
$458$	−5.93929e6	−1.32303
$459$	−4.44460e6	−0.984694
$460$	0	0
$461$	2.09080e6	0.458206	0.229103	−	0.973402i	$-0.426421\pi$
$461$	2.09080e6	0.458206	0.229103	+	0.973402i	$0.426421\pi$
$462$	−7.09909e6	−1.54738
$463$	441416.	0.0956964	0.0478482	−	0.998855i	$-0.484764\pi$
$463$	441416.	0.0956964	0.0478482	+	0.998855i	$0.484764\pi$
$464$	−9.62448e6	−2.07531
$465$	0	0
$466$	1.07244e6	0.228775
$467$	5.52823e6	1.17299	0.586495	−	0.809953i	$-0.300507\pi$
$467$	5.52823e6	1.17299	0.586495	+	0.809953i	$0.300507\pi$
$468$	−1.91015e6	−0.403138
$469$	−7.16548e6	−1.50423
$470$	0	0
$471$	3.73430e6	0.775635
$472$	4.86202e6	1.00453
$473$	2.79583e6	0.574590
$474$	−2.32001e6	−0.474290
$475$	0	0
$476$	3.70734e6	0.749973
$477$	−2.13024e6	−0.428680
$478$	−2.27524e6	−0.455469
$479$	4.55856e6	0.907797	0.453899	−	0.891053i	$-0.350033\pi$
$479$	4.55856e6	0.907797	0.453899	+	0.891053i	$0.350033\pi$
$480$	0	0
$481$	1.03907e7	2.04778
$482$	6.09596e6	1.19516
$483$	7.89005e6	1.53890
$484$	985541.	0.191232
$485$	0	0
$486$	5.71155e6	1.09689
$487$	−2.76910e6	−0.529073	−0.264537	−	0.964376i	$-0.585219\pi$
$487$	−2.76910e6	−0.529073	−0.264537	+	0.964376i	$0.585219\pi$
$488$	−1.71066e6	−0.325173
$489$	−3.30411e6	−0.624860
$490$	0	0
$491$	−6.07096e6	−1.13646	−0.568229	−	0.822870i	$-0.692371\pi$
$491$	−6.07096e6	−1.13646	−0.568229	+	0.822870i	$0.692371\pi$
$492$	−126786.	−0.0236134
$493$	8.33018e6	1.54361
$494$	2.32737e6	0.429089
$495$	0	0
$496$	−1.81874e6	−0.331945
$497$	−1.61950e7	−2.94096
$498$	7.05520e6	1.27478
$499$	−113670.	−0.0204359	−0.0102180	−	0.999948i	$-0.503253\pi$
$499$	−113670.	−0.0204359	−0.0102180	+	0.999948i	$0.503253\pi$
$500$	0	0
$501$	1.81634e6	0.323298
$502$	6.81668e6	1.20730
$503$	−6.19918e6	−1.09248	−0.546241	−	0.837628i	$-0.683942\pi$
$503$	−6.19918e6	−1.09248	−0.546241	+	0.837628i	$0.683942\pi$
$504$	2.52357e6	0.442526
$505$	0	0
$506$	−1.19279e7	−2.07104
$507$	5.24643e6	0.906450
$508$	−3.23235e6	−0.555723
$509$	3.56709e6	0.610267	0.305133	−	0.952310i	$-0.401299\pi$
$509$	3.56709e6	0.610267	0.305133	+	0.952310i	$0.401299\pi$
$510$	0	0
$511$	8.64062e6	1.46384
$512$	2.85601e6	0.481487
$513$	−1.44942e6	−0.243164
$514$	−9.67203e6	−1.61477
$515$	0	0
$516$	−1.11714e6	−0.184707
$517$	−5.18170e6	−0.852600
$518$	1.55579e7	2.54757
$519$	6.39547e6	1.04221
$520$	0	0
$521$	−1.78665e6	−0.288366	−0.144183	−	0.989551i	$-0.546055\pi$
$521$	−1.78665e6	−0.288366	−0.144183	+	0.989551i	$0.546055\pi$
$522$	−6.42635e6	−1.03226
$523$	−1.51608e6	−0.242364	−0.121182	−	0.992630i	$-0.538668\pi$
$523$	−1.51608e6	−0.242364	−0.121182	+	0.992630i	$0.538668\pi$
$524$	−3.13562e6	−0.498878
$525$	0	0
$526$	−3.14345e6	−0.495384
$527$	1.57415e6	0.246900
$528$	6.58429e6	1.02784
$529$	6.82054e6	1.05969
$530$	0	0
$531$	5.64921e6	0.869464
$532$	1.20899e6	0.185201
$533$	−624438.	−0.0952075
$534$	6.09609e6	0.925120
$535$	0	0
$536$	3.81916e6	0.574191
$537$	1.65572e6	0.247771
$538$	414750.	0.0617776
$539$	−1.02969e7	−1.52664
$540$	0	0
$541$	−3.59573e6	−0.528194	−0.264097	−	0.964496i	$-0.585074\pi$
$541$	−3.59573e6	−0.528194	−0.264097	+	0.964496i	$0.585074\pi$
$542$	−1.36390e6	−0.199427
$543$	−2.29458e6	−0.333967
$544$	−6.19145e6	−0.897006
$545$	0	0
$546$	1.39706e7	2.00556
$547$	4.54449e6	0.649407	0.324704	−	0.945816i	$-0.394735\pi$
$547$	4.54449e6	0.649407	0.324704	+	0.945816i	$0.394735\pi$
$548$	4.67172e6	0.664546
$549$	−1.98762e6	−0.281451
$550$	0	0
$551$	2.71652e6	0.381184
$552$	−4.20536e6	−0.587428
$553$	−5.93561e6	−0.825377
$554$	−460936.	−0.0638067
$555$	0	0
$556$	−5.73512e6	−0.786784
$557$	7.11805e6	0.972127	0.486064	−	0.873923i	$-0.338432\pi$
$557$	7.11805e6	0.972127	0.486064	+	0.873923i	$0.338432\pi$
$558$	−1.21439e6	−0.165109
$559$	−5.50205e6	−0.744724
$560$	0	0
$561$	−5.69884e6	−0.764503
$562$	1.89211e6	0.252701
$563$	−3.72584e6	−0.495397	−0.247699	−	0.968837i	$-0.579674\pi$
$563$	−3.72584e6	−0.495397	−0.247699	+	0.968837i	$0.579674\pi$
$564$	2.07046e6	0.274075
$565$	0	0
$566$	−5.61522e6	−0.736759
$567$	−2.86024e6	−0.373633
$568$	8.63184e6	1.12262
$569$	6.90628e6	0.894259	0.447130	−	0.894469i	$-0.352446\pi$
$569$	6.90628e6	0.894259	0.447130	+	0.894469i	$0.352446\pi$
$570$	0	0
$571$	−7.64414e6	−0.981156	−0.490578	−	0.871397i	$-0.663214\pi$
$571$	−7.64414e6	−0.981156	−0.490578	+	0.871397i	$0.663214\pi$
$572$	−7.32748e6	−0.936406
$573$	−9.38809e6	−1.19451
$574$	−934962.	−0.118444
$575$	0	0
$576$	−216794.	−0.0272265
$577$	−8.30903e6	−1.03899	−0.519495	−	0.854474i	$-0.673880\pi$
$577$	−8.30903e6	−1.03899	−0.519495	+	0.854474i	$0.673880\pi$
$578$	−1.36086e6	−0.169431
$579$	−2.52641e6	−0.313190
$580$	0	0
$581$	1.80503e7	2.21842
$582$	−9.91129e6	−1.21289
$583$	−8.17175e6	−0.995735
$584$	−4.60540e6	−0.558773
$585$	0	0
$586$	−3.73260e6	−0.449022
$587$	6.62766e6	0.793899	0.396949	−	0.917841i	$-0.370069\pi$
$587$	6.62766e6	0.793899	0.396949	+	0.917841i	$0.370069\pi$
$588$	4.11437e6	0.490750
$589$	513342.	0.0609703
$590$	0	0
$591$	−9.27056e6	−1.09179
$592$	−1.44297e7	−1.69220
$593$	1.27557e7	1.48960	0.744799	−	0.667289i	$-0.232545\pi$
$593$	1.27557e7	1.48960	0.744799	+	0.667289i	$0.232545\pi$
$594$	1.31531e7	1.52955
$595$	0	0
$596$	−925140.	−0.106682
$597$	7.00552e6	0.804460
$598$	2.34735e7	2.68427
$599$	3.78319e6	0.430815	0.215408	−	0.976524i	$-0.430892\pi$
$599$	3.78319e6	0.430815	0.215408	+	0.976524i	$0.430892\pi$
$600$	0	0
$601$	1.84268e6	0.208096	0.104048	−	0.994572i	$-0.466820\pi$
$601$	1.84268e6	0.208096	0.104048	+	0.994572i	$0.466820\pi$
$602$	−8.23815e6	−0.926485
$603$	4.43751e6	0.496988
$604$	−8.28747e6	−0.924335
$605$	0	0
$606$	4.88426e6	0.540278
$607$	1.54669e7	1.70385	0.851927	−	0.523661i	$-0.175434\pi$
$607$	1.54669e7	1.70385	0.851927	+	0.523661i	$0.175434\pi$
$608$	−2.01907e6	−0.221510
$609$	1.63067e7	1.78165
$610$	0	0
$611$	1.01973e7	1.10505
$612$	−2.29592e6	−0.247787
$613$	−1.50804e6	−0.162092	−0.0810462	−	0.996710i	$-0.525826\pi$
$613$	−1.50804e6	−0.162092	−0.0810462	+	0.996710i	$0.525826\pi$
$614$	8.38048e6	0.897115
$615$	0	0
$616$	9.68058e6	1.02790
$617$	1.07239e7	1.13407	0.567036	−	0.823693i	$-0.308090\pi$
$617$	1.07239e7	1.13407	0.567036	+	0.823693i	$0.308090\pi$
$618$	3.80657e6	0.400925
$619$	240920.	0.0252724	0.0126362	−	0.999920i	$-0.495978\pi$
$619$	240920.	0.0252724	0.0126362	+	0.999920i	$0.495978\pi$
$620$	0	0
$621$	−1.46186e7	−1.52117
$622$	−9.94912e6	−1.03112
$623$	1.55965e7	1.60993
$624$	−1.29575e7	−1.33218
$625$	0	0
$626$	−9.79327e6	−0.998831
$627$	−1.85843e6	−0.188789
$628$	5.77119e6	0.583938
$629$	1.24892e7	1.25866
$630$	0	0
$631$	7.38759e6	0.738634	0.369317	−	0.929303i	$-0.379592\pi$
$631$	7.38759e6	0.738634	0.369317	+	0.929303i	$0.379592\pi$
$632$	3.16365e6	0.315062
$633$	−7.52590e6	−0.746534
$634$	−5.46695e6	−0.540160
$635$	0	0
$636$	3.26521e6	0.320087
$637$	2.02638e7	1.97867
$638$	−2.46519e7	−2.39772
$639$	1.00294e7	0.971677
$640$	0	0
$641$	−7.75177e6	−0.745171	−0.372585	−	0.927998i	$-0.621529\pi$
$641$	−7.75177e6	−0.745171	−0.372585	+	0.927998i	$0.621529\pi$
$642$	−3.28043e6	−0.314118
$643$	9.16769e6	0.874445	0.437222	−	0.899353i	$-0.355962\pi$
$643$	9.16769e6	0.874445	0.437222	+	0.899353i	$0.355962\pi$
$644$	1.21937e7	1.15857
$645$	0	0
$646$	2.79739e6	0.263737
$647$	2.86492e6	0.269061	0.134531	−	0.990909i	$-0.457047\pi$
$647$	2.86492e6	0.269061	0.134531	+	0.990909i	$0.457047\pi$
$648$	1.52450e6	0.142623
$649$	2.16707e7	2.01958
$650$	0	0
$651$	3.08147e6	0.284975
$652$	−5.10636e6	−0.470427
$653$	−368304.	−0.0338005	−0.0169003	−	0.999857i	$-0.505380\pi$
$653$	−368304.	−0.0338005	−0.0169003	+	0.999857i	$0.505380\pi$
$654$	3.18434e6	0.291122
$655$	0	0
$656$	867162.	0.0786757
$657$	−5.35104e6	−0.483643
$658$	1.52683e7	1.37476
$659$	−2.33942e6	−0.209843	−0.104921	−	0.994481i	$-0.533459\pi$
$659$	−2.33942e6	−0.209843	−0.104921	+	0.994481i	$0.533459\pi$
$660$	0	0
$661$	1.01068e7	0.899726	0.449863	−	0.893098i	$-0.351473\pi$
$661$	1.01068e7	0.899726	0.449863	+	0.893098i	$0.351473\pi$
$662$	−2.07087e7	−1.83657
$663$	1.12150e7	0.990869
$664$	−9.62073e6	−0.846814
$665$	0	0
$666$	−9.63483e6	−0.841702
$667$	2.73985e7	2.38458
$668$	2.80707e6	0.243396
$669$	−4.07103e6	−0.351673
$670$	0	0
$671$	−7.62466e6	−0.653753
$672$	−1.21200e7	−1.03533
$673$	−1.58182e7	−1.34623	−0.673117	−	0.739536i	$-0.735045\pi$
$673$	−1.58182e7	−1.34623	−0.673117	+	0.739536i	$0.735045\pi$
$674$	−1.51281e7	−1.28273
$675$	0	0
$676$	8.10812e6	0.682422
$677$	−4.41987e6	−0.370628	−0.185314	−	0.982679i	$-0.559330\pi$
$677$	−4.41987e6	−0.370628	−0.185314	+	0.982679i	$0.559330\pi$
$678$	1.33130e7	1.11225
$679$	−2.53574e7	−2.11072
$680$	0	0
$681$	5.47610e6	0.452484
$682$	−4.65847e6	−0.383515
$683$	1.34927e7	1.10675	0.553373	−	0.832934i	$-0.313341\pi$
$683$	1.34927e7	1.10675	0.553373	+	0.832934i	$0.313341\pi$
$684$	−748714.	−0.0611893
$685$	0	0
$686$	7.16390e6	0.581219
$687$	−9.33317e6	−0.754462
$688$	7.64075e6	0.615410
$689$	1.60816e7	1.29057
$690$	0	0
$691$	−5.85270e6	−0.466295	−0.233148	−	0.972441i	$-0.574903\pi$
$691$	−5.85270e6	−0.466295	−0.233148	+	0.972441i	$0.574903\pi$
$692$	9.88390e6	0.784627
$693$	1.12479e7	0.889690
$694$	1.28219e7	1.01054
$695$	0	0
$696$	−8.69138e6	−0.680088
$697$	−750546.	−0.0585188
$698$	1.28113e7	0.995304
$699$	1.68527e6	0.130459
$700$	0	0
$701$	1.35498e7	1.04144	0.520722	−	0.853726i	$-0.325663\pi$
$701$	1.35498e7	1.04144	0.520722	+	0.853726i	$0.325663\pi$
$702$	−2.58847e7	−1.98244
$703$	4.07280e6	0.310817
$704$	−831636.	−0.0632414
$705$	0	0
$706$	7.07533e6	0.534238
$707$	1.24961e7	0.940213
$708$	−8.65904e6	−0.649212
$709$	−1.92155e6	−0.143561	−0.0717804	−	0.997420i	$-0.522868\pi$
$709$	−1.92155e6	−0.143561	−0.0717804	+	0.997420i	$0.522868\pi$
$710$	0	0
$711$	3.67586e6	0.272700
$712$	−8.31285e6	−0.614540
$713$	5.17750e6	0.381414
$714$	1.67921e7	1.23270
$715$	0	0
$716$	2.55884e6	0.186535
$717$	−3.57539e6	−0.259732
$718$	−1.37945e7	−0.998609
$719$	1.04598e7	0.754576	0.377288	−	0.926096i	$-0.376857\pi$
$719$	1.04598e7	0.754576	0.377288	+	0.926096i	$0.376857\pi$
$720$	0	0
$721$	9.73889e6	0.697704
$722$	912247.	0.0651283
$723$	9.57937e6	0.681540
$724$	−3.54617e6	−0.251427
$725$	0	0
$726$	4.46392e6	0.314322
$727$	−1.28779e7	−0.903671	−0.451836	−	0.892101i	$-0.649231\pi$
$727$	−1.28779e7	−0.903671	−0.451836	+	0.892101i	$0.649231\pi$
$728$	−1.90509e7	−1.33225
$729$	1.25034e7	0.871384
$730$	0	0
$731$	−6.61322e6	−0.457741
$732$	3.04660e6	0.210154
$733$	1.35039e7	0.928324	0.464162	−	0.885750i	$-0.346355\pi$
$733$	1.35039e7	0.928324	0.464162	+	0.885750i	$0.346355\pi$
$734$	2.54497e7	1.74358
$735$	0	0
$736$	−2.03641e7	−1.38571
$737$	1.70226e7	1.15440
$738$	579012.	0.0391333
$739$	−6.14657e6	−0.414020	−0.207010	−	0.978339i	$-0.566373\pi$
$739$	−6.14657e6	−0.414020	−0.207010	+	0.978339i	$0.566373\pi$
$740$	0	0
$741$	3.65729e6	0.244689
$742$	2.40787e7	1.60555
$743$	−7.99529e6	−0.531328	−0.265664	−	0.964066i	$-0.585591\pi$
$743$	−7.99529e6	−0.531328	−0.265664	+	0.964066i	$0.585591\pi$
$744$	−1.64241e6	−0.108780
$745$	0	0
$746$	3.26334e6	0.214692
$747$	−1.11784e7	−0.732955
$748$	−8.80729e6	−0.575557
$749$	−8.39279e6	−0.546641
$750$	0	0
$751$	−7.12326e6	−0.460870	−0.230435	−	0.973088i	$-0.574015\pi$
$751$	−7.12326e6	−0.460870	−0.230435	+	0.973088i	$0.574015\pi$
$752$	−1.41611e7	−0.913171
$753$	1.07119e7	0.688463
$754$	4.85137e7	3.10768
$755$	0	0
$756$	−1.34462e7	−0.855650
$757$	3.12683e7	1.98319	0.991597	−	0.129366i	$-0.0412942\pi$
$757$	3.12683e7	1.98319	0.991597	+	0.129366i	$0.0412942\pi$
$758$	−1.32151e7	−0.835403
$759$	−1.87439e7	−1.18101
$760$	0	0
$761$	−1.91573e7	−1.19914	−0.599572	−	0.800321i	$-0.704663\pi$
$761$	−1.91573e7	−1.19914	−0.599572	+	0.800321i	$0.704663\pi$
$762$	−1.46406e7	−0.913424
$763$	8.14694e6	0.506621
$764$	−1.45089e7	−0.899291
$765$	0	0
$766$	3.17265e7	1.95367
$767$	−4.26469e7	−2.61757
$768$	1.41135e7	0.863436
$769$	−1.24760e7	−0.760782	−0.380391	−	0.924826i	$-0.624211\pi$
$769$	−1.24760e7	−0.760782	−0.380391	+	0.924826i	$0.624211\pi$
$770$	0	0
$771$	−1.51989e7	−0.920823
$772$	−3.90446e6	−0.235786
$773$	135051.	0.00812922	0.00406461	−	0.999992i	$-0.498706\pi$
$773$	135051.	0.00812922	0.00406461	+	0.999992i	$0.498706\pi$
$774$	5.10180e6	0.306105
$775$	0	0
$776$	1.35154e7	0.805702
$777$	2.44481e7	1.45276
$778$	−1.74486e7	−1.03350
$779$	−244758.	−0.0144508
$780$	0	0
$781$	3.84733e7	2.25700
$782$	2.82141e7	1.64987
$783$	−3.02129e7	−1.76111
$784$	−2.81406e7	−1.63509
$785$	0	0
$786$	−1.42025e7	−0.819990
$787$	2.56828e7	1.47810	0.739052	−	0.673648i	$-0.235274\pi$
$787$	2.56828e7	1.47810	0.739052	+	0.673648i	$0.235274\pi$
$788$	−1.43272e7	−0.821952
$789$	−4.93970e6	−0.282493
$790$	0	0
$791$	3.40605e7	1.93557
$792$	−5.99508e6	−0.339611
$793$	1.50049e7	0.847327
$794$	−3.30123e7	−1.85834
$795$	0	0
$796$	1.08267e7	0.605639
$797$	1.33238e6	0.0742987	0.0371494	−	0.999310i	$-0.488172\pi$
$797$	1.33238e6	0.0742987	0.0371494	+	0.999310i	$0.488172\pi$
$798$	5.47601e6	0.304409
$799$	1.22567e7	0.679214
$800$	0	0
$801$	−9.65874e6	−0.531911
$802$	−2.44191e7	−1.34058
$803$	−2.05269e7	−1.12340
$804$	−6.80175e6	−0.371091
$805$	0	0
$806$	9.16763e6	0.497072
$807$	651750.	0.0352287
$808$	−6.66036e6	−0.358897
$809$	−3.18053e6	−0.170855	−0.0854277	−	0.996344i	$-0.527226\pi$
$809$	−3.18053e6	−0.170855	−0.0854277	+	0.996344i	$0.527226\pi$
$810$	0	0
$811$	2.64138e7	1.41019	0.705096	−	0.709112i	$-0.250904\pi$
$811$	2.64138e7	1.41019	0.705096	+	0.709112i	$0.250904\pi$
$812$	2.52012e7	1.34132
$813$	−2.14327e6	−0.113724
$814$	−3.69598e7	−1.95510
$815$	0	0
$816$	−1.55744e7	−0.818814
$817$	−2.15661e6	−0.113036
$818$	−1.59114e7	−0.831426
$819$	−2.21353e7	−1.15312
$820$	0	0
$821$	−3.78773e7	−1.96119	−0.980597	−	0.196032i	$-0.937194\pi$
$821$	−3.78773e7	−1.96119	−0.980597	+	0.196032i	$0.937194\pi$
$822$	2.11601e7	1.09229
$823$	−1.33910e7	−0.689151	−0.344575	−	0.938759i	$-0.611977\pi$
$823$	−1.33910e7	−0.689151	−0.344575	+	0.938759i	$0.611977\pi$
$824$	−5.19078e6	−0.266327
$825$	0	0
$826$	−6.38546e7	−3.25643
$827$	2.00472e7	1.01927	0.509637	−	0.860390i	$-0.329780\pi$
$827$	2.00472e7	1.01927	0.509637	+	0.860390i	$0.329780\pi$
$828$	−7.55143e6	−0.382784
$829$	−9.53248e6	−0.481748	−0.240874	−	0.970556i	$-0.577434\pi$
$829$	−9.53248e6	−0.481748	−0.240874	+	0.970556i	$0.577434\pi$
$830$	0	0
$831$	−724328.	−0.0363859
$832$	1.63662e6	0.0819669
$833$	2.43562e7	1.21618
$834$	−2.59767e7	−1.29321
$835$	0	0
$836$	−2.87212e6	−0.142130
$837$	−5.70933e6	−0.281690
$838$	2.93162e7	1.44211
$839$	−1.60559e7	−0.787460	−0.393730	−	0.919226i	$-0.628816\pi$
$839$	−1.60559e7	−0.787460	−0.393730	+	0.919226i	$0.628816\pi$
$840$	0	0
$841$	3.61145e7	1.76072
$842$	−7.71416e6	−0.374980
$843$	2.97332e6	0.144103
$844$	−1.16309e7	−0.562029
$845$	0	0
$846$	−9.45549e6	−0.454211
$847$	1.14207e7	0.546995
$848$	−2.23326e7	−1.06647
$849$	−8.82391e6	−0.420138
$850$	0	0
$851$	4.10778e7	1.94439
$852$	−1.53729e7	−0.725532
$853$	3.42215e7	1.61037	0.805187	−	0.593021i	$-0.202065\pi$
$853$	3.42215e7	1.61037	0.805187	+	0.593021i	$0.202065\pi$
$854$	2.24667e7	1.05413
$855$	0	0
$856$	4.47332e6	0.208663
$857$	−9.61977e6	−0.447417	−0.223709	−	0.974656i	$-0.571816\pi$
$857$	−9.61977e6	−0.447417	−0.223709	+	0.974656i	$0.571816\pi$
$858$	−3.31892e7	−1.53914
$859$	2.07271e7	0.958421	0.479211	−	0.877700i	$-0.340923\pi$
$859$	2.07271e7	0.958421	0.479211	+	0.877700i	$0.340923\pi$
$860$	0	0
$861$	−1.46923e6	−0.0675431
$862$	487284.	0.0223364
$863$	−2.60906e6	−0.119250	−0.0596249	−	0.998221i	$-0.518990\pi$
$863$	−2.60906e6	−0.119250	−0.0596249	+	0.998221i	$0.518990\pi$
$864$	2.24559e7	1.02340
$865$	0	0
$866$	−3.14596e7	−1.42547
$867$	−2.13849e6	−0.0966182
$868$	4.76228e6	0.214544
$869$	1.41008e7	0.633426
$870$	0	0
$871$	−3.34995e7	−1.49621
$872$	−4.34228e6	−0.193386
$873$	1.57036e7	0.697370
$874$	9.20081e6	0.407425
$875$	0	0
$876$	8.20201e6	0.361127
$877$	−278553.	−0.0122295	−0.00611475	−	0.999981i	$-0.501946\pi$
$877$	−278553.	−0.0122295	−0.00611475	+	0.999981i	$0.501946\pi$
$878$	−1.13337e6	−0.0496176
$879$	−5.86552e6	−0.256056
$880$	0	0
$881$	−2.88926e7	−1.25414	−0.627071	−	0.778962i	$-0.715746\pi$
$881$	−2.88926e7	−1.25414	−0.627071	+	0.778962i	$0.715746\pi$
$882$	−1.87897e7	−0.813296
$883$	−2.89440e7	−1.24927	−0.624636	−	0.780916i	$-0.714753\pi$
$883$	−2.89440e7	−1.24927	−0.624636	+	0.780916i	$0.714753\pi$
$884$	1.73323e7	0.745977
$885$	0	0
$886$	−5.62537e7	−2.40750
$887$	1.84363e7	0.786802	0.393401	−	0.919367i	$-0.371298\pi$
$887$	1.84363e7	0.786802	0.393401	+	0.919367i	$0.371298\pi$
$888$	−1.30307e7	−0.554544
$889$	−3.74572e7	−1.58957
$890$	0	0
$891$	6.79489e6	0.286740
$892$	−6.29160e6	−0.264758
$893$	3.99699e6	0.167728
$894$	−4.19034e6	−0.175350
$895$	0	0
$896$	3.77088e7	1.56918
$897$	3.68870e7	1.53071
$898$	5.73620e7	2.37374
$899$	1.07006e7	0.441577
$900$	0	0
$901$	1.93293e7	0.793240
$902$	2.22113e6	0.0908986
$903$	−1.29457e7	−0.528329
$904$	−1.81541e7	−0.738845
$905$	0	0
$906$	−3.75373e7	−1.51930
$907$	−8.77058e6	−0.354006	−0.177003	−	0.984210i	$-0.556640\pi$
$907$	−8.77058e6	−0.354006	−0.177003	+	0.984210i	$0.556640\pi$
$908$	8.46306e6	0.340653
$909$	−7.73870e6	−0.310641
$910$	0	0
$911$	1.15811e7	0.462332	0.231166	−	0.972914i	$-0.425746\pi$
$911$	1.15811e7	0.462332	0.231166	+	0.972914i	$0.425746\pi$
$912$	−5.07891e6	−0.202201
$913$	−4.28810e7	−1.70250
$914$	−9.77426e6	−0.387007
$915$	0	0
$916$	−1.44240e7	−0.567998
$917$	−3.63363e7	−1.42698
$918$	−3.11122e7	−1.21850
$919$	1.32806e7	0.518713	0.259357	−	0.965782i	$-0.416489\pi$
$919$	1.32806e7	0.518713	0.259357	+	0.965782i	$0.416489\pi$
$920$	0	0
$921$	1.31693e7	0.511581
$922$	1.46356e7	0.567001
$923$	−7.57136e7	−2.92529
$924$	−1.72407e7	−0.664315
$925$	0	0
$926$	3.08991e6	0.118418
$927$	−6.03119e6	−0.230517
$928$	−4.20873e7	−1.60428
$929$	1.88206e7	0.715475	0.357737	−	0.933822i	$-0.383548\pi$
$929$	1.88206e7	0.715475	0.357737	+	0.933822i	$0.383548\pi$
$930$	0	0
$931$	7.94272e6	0.300328
$932$	2.60450e6	0.0982166
$933$	−1.56343e7	−0.587997
$934$	3.86976e7	1.45150
$935$	0	0
$936$	1.17980e7	0.440169
$937$	−5.62110e6	−0.209157	−0.104579	−	0.994517i	$-0.533349\pi$
$937$	−5.62110e6	−0.209157	−0.104579	+	0.994517i	$0.533349\pi$
$938$	−5.01584e7	−1.86139
$939$	−1.53894e7	−0.569585
$940$	0	0
$941$	−3.34570e7	−1.23172	−0.615861	−	0.787855i	$-0.711192\pi$
$941$	−3.34570e7	−1.23172	−0.615861	+	0.787855i	$0.711192\pi$
$942$	2.61401e7	0.959799
$943$	−2.46860e6	−0.0904006
$944$	5.92241e7	2.16306
$945$	0	0
$946$	1.95708e7	0.711019
$947$	7.56871e6	0.274250	0.137125	−	0.990554i	$-0.456214\pi$
$947$	7.56871e6	0.274250	0.137125	+	0.990554i	$0.456214\pi$
$948$	−5.63431e6	−0.203620
$949$	4.03960e7	1.45604
$950$	0	0
$951$	−8.59092e6	−0.308027
$952$	−2.28983e7	−0.818862
$953$	3.39295e7	1.21017	0.605083	−	0.796162i	$-0.293140\pi$
$953$	3.39295e7	1.21017	0.605083	+	0.796162i	$0.293140\pi$
$954$	−1.49117e7	−0.530464
$955$	0	0
$956$	−5.52560e6	−0.195539
$957$	−3.87387e7	−1.36730
$958$	3.19099e7	1.12334
$959$	5.41370e7	1.90085
$960$	0	0
$961$	−2.66071e7	−0.929370
$962$	7.27351e7	2.53400
$963$	5.19757e6	0.180607
$964$	1.48045e7	0.513098
$965$	0	0
$966$	5.52303e7	1.90430
$967$	2.13057e6	0.0732706	0.0366353	−	0.999329i	$-0.488336\pi$
$967$	2.13057e6	0.0732706	0.0366353	+	0.999329i	$0.488336\pi$
$968$	−6.08716e6	−0.208798
$969$	4.39590e6	0.150397
$970$	0	0
$971$	−5.18977e7	−1.76644	−0.883222	−	0.468956i	$-0.844630\pi$
$971$	−5.18977e7	−1.76644	−0.883222	+	0.468956i	$0.844630\pi$
$972$	1.38709e7	0.470912
$973$	−6.64599e7	−2.25049
$974$	−1.93837e7	−0.654695
$975$	0	0
$976$	−2.08375e7	−0.700197
$977$	−2.16360e7	−0.725172	−0.362586	−	0.931950i	$-0.618106\pi$
$977$	−2.16360e7	−0.725172	−0.362586	+	0.931950i	$0.618106\pi$
$978$	−2.31288e7	−0.773225
$979$	−3.70516e7	−1.23552
$980$	0	0
$981$	−5.04531e6	−0.167385
$982$	−4.24967e7	−1.40630
$983$	−1.76106e7	−0.581287	−0.290644	−	0.956831i	$-0.593869\pi$
$983$	−1.76106e7	−0.581287	−0.290644	+	0.956831i	$0.593869\pi$
$984$	783090.	0.0257824
$985$	0	0
$986$	5.83112e7	1.91012
$987$	2.39930e7	0.783957
$988$	5.65218e6	0.184214
$989$	−2.17513e7	−0.707123
$990$	0	0
$991$	3.81030e6	0.123247	0.0616233	−	0.998099i	$-0.480372\pi$
$991$	3.81030e6	0.123247	0.0616233	+	0.998099i	$0.480372\pi$
$992$	−7.95325e6	−0.256605
$993$	−3.25422e7	−1.04731
$994$	−1.13365e8	−3.63925
$995$	0	0
$996$	1.71341e7	0.547283
$997$	−5.67513e7	−1.80816	−0.904082	−	0.427359i	$-0.859444\pi$
$997$	−5.67513e7	−1.80816	−0.904082	+	0.427359i	$0.859444\pi$
$998$	−795690.	−0.0252882
$999$	−4.52972e7	−1.43601

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.6.a.c.1.1		1
5.4	even	2		95.6.a.a.1.1	✓	1
15.14	odd	2		855.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.6.a.a.1.1	✓	1	5.4	even	2
475.6.a.c.1.1		1	1.1	even	1	trivial
855.6.a.b.1.1		1	15.14	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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