LMFDB - Embedded newform 378.6.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,6,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, 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("378.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	378.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:	$60.6250838893$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} +91.0000 q^{5} -49.0000 q^{7} +64.0000 q^{8} +O(q^{10})$

\(q+4.00000 q^{2} +16.0000 q^{4} +91.0000 q^{5} -49.0000 q^{7} +64.0000 q^{8} +364.000 q^{10} -61.0000 q^{11} +156.000 q^{13} -196.000 q^{14} +256.000 q^{16} +614.000 q^{17} +2207.00 q^{19} +1456.00 q^{20} -244.000 q^{22} -3139.00 q^{23} +5156.00 q^{25} +624.000 q^{26} -784.000 q^{28} +3424.00 q^{29} +6435.00 q^{31} +1024.00 q^{32} +2456.00 q^{34} -4459.00 q^{35} -6199.00 q^{37} +8828.00 q^{38} +5824.00 q^{40} +4929.00 q^{41} -4222.00 q^{43} -976.000 q^{44} -12556.0 q^{46} +5142.00 q^{47} +2401.00 q^{49} +20624.0 q^{50} +2496.00 q^{52} +5724.00 q^{53} -5551.00 q^{55} -3136.00 q^{56} +13696.0 q^{58} -15902.0 q^{59} +18624.0 q^{61} +25740.0 q^{62} +4096.00 q^{64} +14196.0 q^{65} +11884.0 q^{67} +9824.00 q^{68} -17836.0 q^{70} -55879.0 q^{71} -42494.0 q^{73} -24796.0 q^{74} +35312.0 q^{76} +2989.00 q^{77} -23622.0 q^{79} +23296.0 q^{80} +19716.0 q^{82} +79400.0 q^{83} +55874.0 q^{85} -16888.0 q^{86} -3904.00 q^{88} +12201.0 q^{89} -7644.00 q^{91} -50224.0 q^{92} +20568.0 q^{94} +200837. q^{95} +104080. q^{97} +9604.00 q^{98} +O(q^{100})\)

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
$5$	91.0000	1.62786	0.813929	−	0.580965i	$-0.197325\pi$
$5$	91.0000	1.62786	0.813929	+	0.580965i	$0.197325\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	64.0000	0.353553
$9$	0	0
$10$	364.000	1.15107
$11$	−61.0000	−0.152002	−0.0760008	−	0.997108i	$-0.524215\pi$
$11$	−61.0000	−0.152002	−0.0760008	+	0.997108i	$0.524215\pi$
$12$	0	0
$13$	156.000	0.256015	0.128008	−	0.991773i	$-0.459142\pi$
$13$	156.000	0.256015	0.128008	+	0.991773i	$0.459142\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	614.000	0.515283	0.257642	−	0.966241i	$-0.417055\pi$
$17$	614.000	0.515283	0.257642	+	0.966241i	$0.417055\pi$
$18$	0	0
$19$	2207.00	1.40255	0.701275	−	0.712891i	$-0.252615\pi$
$19$	2207.00	1.40255	0.701275	+	0.712891i	$0.252615\pi$
$20$	1456.00	0.813929
$21$	0	0
$22$	−244.000	−0.107481
$23$	−3139.00	−1.23729	−0.618645	−	0.785670i	$-0.712318\pi$
$23$	−3139.00	−1.23729	−0.618645	+	0.785670i	$0.712318\pi$
$24$	0	0
$25$	5156.00	1.64992
$26$	624.000	0.181030
$27$	0	0
$28$	−784.000	−0.188982
$29$	3424.00	0.756030	0.378015	−	0.925800i	$-0.376607\pi$
$29$	3424.00	0.756030	0.378015	+	0.925800i	$0.376607\pi$
$30$	0	0
$31$	6435.00	1.20266	0.601332	−	0.798999i	$-0.294637\pi$
$31$	6435.00	1.20266	0.601332	+	0.798999i	$0.294637\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	2456.00	0.364360
$35$	−4459.00	−0.615272
$36$	0	0
$37$	−6199.00	−0.744419	−0.372209	−	0.928149i	$-0.621400\pi$
$37$	−6199.00	−0.744419	−0.372209	+	0.928149i	$0.621400\pi$
$38$	8828.00	0.991753
$39$	0	0
$40$	5824.00	0.575535
$41$	4929.00	0.457930	0.228965	−	0.973435i	$-0.426466\pi$
$41$	4929.00	0.457930	0.228965	+	0.973435i	$0.426466\pi$
$42$	0	0
$43$	−4222.00	−0.348215	−0.174107	−	0.984727i	$-0.555704\pi$
$43$	−4222.00	−0.348215	−0.174107	+	0.984727i	$0.555704\pi$
$44$	−976.000	−0.0760008
$45$	0	0
$46$	−12556.0	−0.874897
$47$	5142.00	0.339537	0.169769	−	0.985484i	$-0.445698\pi$
$47$	5142.00	0.339537	0.169769	+	0.985484i	$0.445698\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	20624.0	1.16667
$51$	0	0
$52$	2496.00	0.128008
$53$	5724.00	0.279905	0.139952	−	0.990158i	$-0.455305\pi$
$53$	5724.00	0.279905	0.139952	+	0.990158i	$0.455305\pi$
$54$	0	0
$55$	−5551.00	−0.247437
$56$	−3136.00	−0.133631
$57$	0	0
$58$	13696.0	0.534594
$59$	−15902.0	−0.594733	−0.297366	−	0.954763i	$-0.596108\pi$
$59$	−15902.0	−0.594733	−0.297366	+	0.954763i	$0.596108\pi$
$60$	0	0
$61$	18624.0	0.640838	0.320419	−	0.947276i	$-0.396176\pi$
$61$	18624.0	0.640838	0.320419	+	0.947276i	$0.396176\pi$
$62$	25740.0	0.850412
$63$	0	0
$64$	4096.00	0.125000
$65$	14196.0	0.416757
$66$	0	0
$67$	11884.0	0.323427	0.161713	−	0.986838i	$-0.448298\pi$
$67$	11884.0	0.323427	0.161713	+	0.986838i	$0.448298\pi$
$68$	9824.00	0.257642
$69$	0	0
$70$	−17836.0	−0.435063
$71$	−55879.0	−1.31554	−0.657768	−	0.753221i	$-0.728499\pi$
$71$	−55879.0	−1.31554	−0.657768	+	0.753221i	$0.728499\pi$
$72$	0	0
$73$	−42494.0	−0.933298	−0.466649	−	0.884443i	$-0.654539\pi$
$73$	−42494.0	−0.933298	−0.466649	+	0.884443i	$0.654539\pi$
$74$	−24796.0	−0.526384
$75$	0	0
$76$	35312.0	0.701275
$77$	2989.00	0.0574512
$78$	0	0
$79$	−23622.0	−0.425842	−0.212921	−	0.977069i	$-0.568298\pi$
$79$	−23622.0	−0.425842	−0.212921	+	0.977069i	$0.568298\pi$
$80$	23296.0	0.406964
$81$	0	0
$82$	19716.0	0.323805
$83$	79400.0	1.26510	0.632551	−	0.774519i	$-0.282008\pi$
$83$	79400.0	1.26510	0.632551	+	0.774519i	$0.282008\pi$
$84$	0	0
$85$	55874.0	0.838808
$86$	−16888.0	−0.246225
$87$	0	0
$88$	−3904.00	−0.0537407
$89$	12201.0	0.163275	0.0816376	−	0.996662i	$-0.473985\pi$
$89$	12201.0	0.163275	0.0816376	+	0.996662i	$0.473985\pi$
$90$	0	0
$91$	−7644.00	−0.0967648
$92$	−50224.0	−0.618645
$93$	0	0
$94$	20568.0	0.240089
$95$	200837.	2.28315
$96$	0	0
$97$	104080.	1.12315	0.561575	−	0.827426i	$-0.310196\pi$
$97$	104080.	1.12315	0.561575	+	0.827426i	$0.310196\pi$
$98$	9604.00	0.101015
$99$	0	0
$100$	82496.0	0.824960
$101$	190102.	1.85431	0.927157	−	0.374673i	$-0.122245\pi$
$101$	190102.	1.85431	0.927157	+	0.374673i	$0.122245\pi$
$102$	0	0
$103$	142211.	1.32081	0.660405	−	0.750910i	$-0.270385\pi$
$103$	142211.	1.32081	0.660405	+	0.750910i	$0.270385\pi$
$104$	9984.00	0.0905151
$105$	0	0
$106$	22896.0	0.197922
$107$	57972.0	0.489507	0.244753	−	0.969585i	$-0.421293\pi$
$107$	57972.0	0.489507	0.244753	+	0.969585i	$0.421293\pi$
$108$	0	0
$109$	−100181.	−0.807642	−0.403821	−	0.914838i	$-0.632318\pi$
$109$	−100181.	−0.807642	−0.403821	+	0.914838i	$0.632318\pi$
$110$	−22204.0	−0.174964
$111$	0	0
$112$	−12544.0	−0.0944911
$113$	206026.	1.51784	0.758920	−	0.651184i	$-0.225727\pi$
$113$	206026.	1.51784	0.758920	+	0.651184i	$0.225727\pi$
$114$	0	0
$115$	−285649.	−2.01413
$116$	54784.0	0.378015
$117$	0	0
$118$	−63608.0	−0.420539
$119$	−30086.0	−0.194759
$120$	0	0
$121$	−157330.	−0.976896
$122$	74496.0	0.453141
$123$	0	0
$124$	102960.	0.601332
$125$	184821.	1.05798
$126$	0	0
$127$	205668.	1.13151	0.565754	−	0.824574i	$-0.308585\pi$
$127$	205668.	1.13151	0.565754	+	0.824574i	$0.308585\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	56784.0	0.294691
$131$	−202250.	−1.02970	−0.514850	−	0.857281i	$-0.672152\pi$
$131$	−202250.	−1.02970	−0.514850	+	0.857281i	$0.672152\pi$
$132$	0	0
$133$	−108143.	−0.530114
$134$	47536.0	0.228697
$135$	0	0
$136$	39296.0	0.182180
$137$	282868.	1.28760	0.643802	−	0.765192i	$-0.277356\pi$
$137$	282868.	1.28760	0.643802	+	0.765192i	$0.277356\pi$
$138$	0	0
$139$	−376796.	−1.65413	−0.827064	−	0.562107i	$-0.809991\pi$
$139$	−376796.	−1.65413	−0.827064	+	0.562107i	$0.809991\pi$
$140$	−71344.0	−0.307636
$141$	0	0
$142$	−223516.	−0.930224
$143$	−9516.00	−0.0389148
$144$	0	0
$145$	311584.	1.23071
$146$	−169976.	−0.659942
$147$	0	0
$148$	−99184.0	−0.372209
$149$	−41574.0	−0.153411	−0.0767054	−	0.997054i	$-0.524440\pi$
$149$	−41574.0	−0.153411	−0.0767054	+	0.997054i	$0.524440\pi$
$150$	0	0
$151$	−552182.	−1.97079	−0.985394	−	0.170290i	$-0.945529\pi$
$151$	−552182.	−1.97079	−0.985394	+	0.170290i	$0.945529\pi$
$152$	141248.	0.495876
$153$	0	0
$154$	11956.0	0.0406241
$155$	585585.	1.95777
$156$	0	0
$157$	−323692.	−1.04805	−0.524026	−	0.851702i	$-0.675571\pi$
$157$	−323692.	−1.04805	−0.524026	+	0.851702i	$0.675571\pi$
$158$	−94488.0	−0.301116
$159$	0	0
$160$	93184.0	0.287767
$161$	153811.	0.467652
$162$	0	0
$163$	−278416.	−0.820777	−0.410389	−	0.911911i	$-0.634607\pi$
$163$	−278416.	−0.820777	−0.410389	+	0.911911i	$0.634607\pi$
$164$	78864.0	0.228965
$165$	0	0
$166$	317600.	0.894562
$167$	−503066.	−1.39583	−0.697917	−	0.716179i	$-0.745890\pi$
$167$	−503066.	−1.39583	−0.697917	+	0.716179i	$0.745890\pi$
$168$	0	0
$169$	−346957.	−0.934456
$170$	223496.	0.593127
$171$	0	0
$172$	−67552.0	−0.174107
$173$	470029.	1.19401	0.597007	−	0.802236i	$-0.296356\pi$
$173$	470029.	1.19401	0.597007	+	0.802236i	$0.296356\pi$
$174$	0	0
$175$	−252644.	−0.623611
$176$	−15616.0	−0.0380004
$177$	0	0
$178$	48804.0	0.115453
$179$	−708456.	−1.65265	−0.826324	−	0.563195i	$-0.809572\pi$
$179$	−708456.	−1.65265	−0.826324	+	0.563195i	$0.809572\pi$
$180$	0	0
$181$	585378.	1.32813	0.664064	−	0.747676i	$-0.268830\pi$
$181$	585378.	1.32813	0.664064	+	0.747676i	$0.268830\pi$
$182$	−30576.0	−0.0684230
$183$	0	0
$184$	−200896.	−0.437448
$185$	−564109.	−1.21181
$186$	0	0
$187$	−37454.0	−0.0783239
$188$	82272.0	0.169769
$189$	0	0
$190$	803348.	1.61443
$191$	−160881.	−0.319096	−0.159548	−	0.987190i	$-0.551004\pi$
$191$	−160881.	−0.319096	−0.159548	+	0.987190i	$0.551004\pi$
$192$	0	0
$193$	−828802.	−1.60161	−0.800806	−	0.598923i	$-0.795595\pi$
$193$	−828802.	−1.60161	−0.800806	+	0.598923i	$0.795595\pi$
$194$	416320.	0.794187
$195$	0	0
$196$	38416.0	0.0714286
$197$	531242.	0.975274	0.487637	−	0.873046i	$-0.337859\pi$
$197$	531242.	0.975274	0.487637	+	0.873046i	$0.337859\pi$
$198$	0	0
$199$	527519.	0.944290	0.472145	−	0.881521i	$-0.343480\pi$
$199$	527519.	0.944290	0.472145	+	0.881521i	$0.343480\pi$
$200$	329984.	0.583335
$201$	0	0
$202$	760408.	1.31120
$203$	−167776.	−0.285752
$204$	0	0
$205$	448539.	0.745445
$206$	568844.	0.933953
$207$	0	0
$208$	39936.0	0.0640039
$209$	−134627.	−0.213190
$210$	0	0
$211$	535970.	0.828771	0.414385	−	0.910102i	$-0.363997\pi$
$211$	535970.	0.828771	0.414385	+	0.910102i	$0.363997\pi$
$212$	91584.0	0.139952
$213$	0	0
$214$	231888.	0.346134
$215$	−384202.	−0.566844
$216$	0	0
$217$	−315315.	−0.454564
$218$	−400724.	−0.571089
$219$	0	0
$220$	−88816.0	−0.123718
$221$	95784.0	0.131920
$222$	0	0
$223$	43385.0	0.0584221	0.0292111	−	0.999573i	$-0.490701\pi$
$223$	43385.0	0.0584221	0.0292111	+	0.999573i	$0.490701\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	824104.	1.07327
$227$	−1.21268e6	−1.56200	−0.781000	−	0.624531i	$-0.785290\pi$
$227$	−1.21268e6	−1.56200	−0.781000	+	0.624531i	$0.785290\pi$
$228$	0	0
$229$	568828.	0.716791	0.358395	−	0.933570i	$-0.383324\pi$
$229$	568828.	0.716791	0.358395	+	0.933570i	$0.383324\pi$
$230$	−1.14260e6	−1.42421
$231$	0	0
$232$	219136.	0.267297
$233$	−353470.	−0.426543	−0.213271	−	0.976993i	$-0.568412\pi$
$233$	−353470.	−0.426543	−0.213271	+	0.976993i	$0.568412\pi$
$234$	0	0
$235$	467922.	0.552718
$236$	−254432.	−0.297366
$237$	0	0
$238$	−120344.	−0.137715
$239$	−1.07506e6	−1.21741	−0.608704	−	0.793397i	$-0.708310\pi$
$239$	−1.07506e6	−1.21741	−0.608704	+	0.793397i	$0.708310\pi$
$240$	0	0
$241$	−1.50741e6	−1.67182	−0.835911	−	0.548865i	$-0.815060\pi$
$241$	−1.50741e6	−1.67182	−0.835911	+	0.548865i	$0.815060\pi$
$242$	−629320.	−0.690769
$243$	0	0
$244$	297984.	0.320419
$245$	218491.	0.232551
$246$	0	0
$247$	344292.	0.359074
$248$	411840.	0.425206
$249$	0	0
$250$	739284.	0.748103
$251$	50040.0	0.0501341	0.0250670	−	0.999686i	$-0.492020\pi$
$251$	50040.0	0.0501341	0.0250670	+	0.999686i	$0.492020\pi$
$252$	0	0
$253$	191479.	0.188070
$254$	822672.	0.800097
$255$	0	0
$256$	65536.0	0.0625000
$257$	−958947.	−0.905653	−0.452826	−	0.891599i	$-0.649584\pi$
$257$	−958947.	−0.905653	−0.452826	+	0.891599i	$0.649584\pi$
$258$	0	0
$259$	303751.	0.281364
$260$	227136.	0.208378
$261$	0	0
$262$	−809000.	−0.728107
$263$	1.70706e6	1.52180	0.760902	−	0.648867i	$-0.224757\pi$
$263$	1.70706e6	1.52180	0.760902	+	0.648867i	$0.224757\pi$
$264$	0	0
$265$	520884.	0.455645
$266$	−432572.	−0.374847
$267$	0	0
$268$	190144.	0.161713
$269$	353479.	0.297840	0.148920	−	0.988849i	$-0.452420\pi$
$269$	353479.	0.297840	0.148920	+	0.988849i	$0.452420\pi$
$270$	0	0
$271$	−2.27774e6	−1.88400	−0.942002	−	0.335608i	$-0.891058\pi$
$271$	−2.27774e6	−1.88400	−0.942002	+	0.335608i	$0.891058\pi$
$272$	157184.	0.128821
$273$	0	0
$274$	1.13147e6	0.910474
$275$	−314516.	−0.250790
$276$	0	0
$277$	−1.42376e6	−1.11490	−0.557451	−	0.830210i	$-0.688221\pi$
$277$	−1.42376e6	−1.11490	−0.557451	+	0.830210i	$0.688221\pi$
$278$	−1.50718e6	−1.16965
$279$	0	0
$280$	−285376.	−0.217532
$281$	1.15324e6	0.871271	0.435635	−	0.900123i	$-0.356524\pi$
$281$	1.15324e6	0.871271	0.435635	+	0.900123i	$0.356524\pi$
$282$	0	0
$283$	−75980.0	−0.0563940	−0.0281970	−	0.999602i	$-0.508977\pi$
$283$	−75980.0	−0.0563940	−0.0281970	+	0.999602i	$0.508977\pi$
$284$	−894064.	−0.657768
$285$	0	0
$286$	−38064.0	−0.0275169
$287$	−241521.	−0.173081
$288$	0	0
$289$	−1.04286e6	−0.734483
$290$	1.24634e6	0.870242
$291$	0	0
$292$	−679904.	−0.466649
$293$	−1.94248e6	−1.32187	−0.660934	−	0.750444i	$-0.729840\pi$
$293$	−1.94248e6	−1.32187	−0.660934	+	0.750444i	$0.729840\pi$
$294$	0	0
$295$	−1.44708e6	−0.968140
$296$	−396736.	−0.263192
$297$	0	0
$298$	−166296.	−0.108478
$299$	−489684.	−0.316766
$300$	0	0
$301$	206878.	0.131613
$302$	−2.20873e6	−1.39356
$303$	0	0
$304$	564992.	0.350637
$305$	1.69478e6	1.04319
$306$	0	0
$307$	−1.78930e6	−1.08352	−0.541761	−	0.840533i	$-0.682242\pi$
$307$	−1.78930e6	−1.08352	−0.541761	+	0.840533i	$0.682242\pi$
$308$	47824.0	0.0287256
$309$	0	0
$310$	2.34234e6	1.38435
$311$	−3.29078e6	−1.92929	−0.964646	−	0.263550i	$-0.915107\pi$
$311$	−3.29078e6	−1.92929	−0.964646	+	0.263550i	$0.915107\pi$
$312$	0	0
$313$	1.28470e6	0.741209	0.370605	−	0.928791i	$-0.379150\pi$
$313$	1.28470e6	0.741209	0.370605	+	0.928791i	$0.379150\pi$
$314$	−1.29477e6	−0.741085
$315$	0	0
$316$	−377952.	−0.212921
$317$	−1.80779e6	−1.01042	−0.505209	−	0.862997i	$-0.668584\pi$
$317$	−1.80779e6	−1.01042	−0.505209	+	0.862997i	$0.668584\pi$
$318$	0	0
$319$	−208864.	−0.114918
$320$	372736.	0.203482
$321$	0	0
$322$	615244.	0.330680
$323$	1.35510e6	0.722711
$324$	0	0
$325$	804336.	0.422405
$326$	−1.11366e6	−0.580377
$327$	0	0
$328$	315456.	0.161903
$329$	−251958.	−0.128333
$330$	0	0
$331$	2.18148e6	1.09441	0.547206	−	0.836998i	$-0.315692\pi$
$331$	2.18148e6	1.09441	0.547206	+	0.836998i	$0.315692\pi$
$332$	1.27040e6	0.632551
$333$	0	0
$334$	−2.01226e6	−0.987004
$335$	1.08144e6	0.526492
$336$	0	0
$337$	−463293.	−0.222219	−0.111109	−	0.993808i	$-0.535440\pi$
$337$	−463293.	−0.222219	−0.111109	+	0.993808i	$0.535440\pi$
$338$	−1.38783e6	−0.660760
$339$	0	0
$340$	893984.	0.419404
$341$	−392535.	−0.182807
$342$	0	0
$343$	−117649.	−0.0539949
$344$	−270208.	−0.123112
$345$	0	0
$346$	1.88012e6	0.844295
$347$	1.63248e6	0.727822	0.363911	−	0.931434i	$-0.381441\pi$
$347$	1.63248e6	0.727822	0.363911	+	0.931434i	$0.381441\pi$
$348$	0	0
$349$	3.46267e6	1.52176	0.760882	−	0.648890i	$-0.224766\pi$
$349$	3.46267e6	1.52176	0.760882	+	0.648890i	$0.224766\pi$
$350$	−1.01058e6	−0.440960
$351$	0	0
$352$	−62464.0	−0.0268703
$353$	−1.07462e6	−0.459006	−0.229503	−	0.973308i	$-0.573710\pi$
$353$	−1.07462e6	−0.459006	−0.229503	+	0.973308i	$0.573710\pi$
$354$	0	0
$355$	−5.08499e6	−2.14151
$356$	195216.	0.0816376
$357$	0	0
$358$	−2.83382e6	−1.16860
$359$	−1.45320e6	−0.595097	−0.297549	−	0.954707i	$-0.596169\pi$
$359$	−1.45320e6	−0.595097	−0.297549	+	0.954707i	$0.596169\pi$
$360$	0	0
$361$	2.39475e6	0.967146
$362$	2.34151e6	0.939128
$363$	0	0
$364$	−122304.	−0.0483824
$365$	−3.86695e6	−1.51928
$366$	0	0
$367$	−2.04242e6	−0.791555	−0.395777	−	0.918346i	$-0.629525\pi$
$367$	−2.04242e6	−0.791555	−0.395777	+	0.918346i	$0.629525\pi$
$368$	−803584.	−0.309323
$369$	0	0
$370$	−2.25644e6	−0.856877
$371$	−280476.	−0.105794
$372$	0	0
$373$	2.67184e6	0.994349	0.497175	−	0.867651i	$-0.334371\pi$
$373$	2.67184e6	0.994349	0.497175	+	0.867651i	$0.334371\pi$
$374$	−149816.	−0.0553833
$375$	0	0
$376$	329088.	0.120045
$377$	534144.	0.193555
$378$	0	0
$379$	2.09525e6	0.749268	0.374634	−	0.927173i	$-0.377768\pi$
$379$	2.09525e6	0.749268	0.374634	+	0.927173i	$0.377768\pi$
$380$	3.21339e6	1.14158
$381$	0	0
$382$	−643524.	−0.225635
$383$	718154.	0.250162	0.125081	−	0.992147i	$-0.460081\pi$
$383$	718154.	0.250162	0.125081	+	0.992147i	$0.460081\pi$
$384$	0	0
$385$	271999.	0.0935224
$386$	−3.31521e6	−1.13251
$387$	0	0
$388$	1.66528e6	0.561575
$389$	2.49082e6	0.834581	0.417290	−	0.908773i	$-0.362980\pi$
$389$	2.49082e6	0.834581	0.417290	+	0.908773i	$0.362980\pi$
$390$	0	0
$391$	−1.92735e6	−0.637555
$392$	153664.	0.0505076
$393$	0	0
$394$	2.12497e6	0.689623
$395$	−2.14960e6	−0.693211
$396$	0	0
$397$	−5.18576e6	−1.65134	−0.825669	−	0.564155i	$-0.809202\pi$
$397$	−5.18576e6	−1.65134	−0.825669	+	0.564155i	$0.809202\pi$
$398$	2.11008e6	0.667714
$399$	0	0
$400$	1.31994e6	0.412480
$401$	−2.05462e6	−0.638072	−0.319036	−	0.947743i	$-0.603359\pi$
$401$	−2.05462e6	−0.638072	−0.319036	+	0.947743i	$0.603359\pi$
$402$	0	0
$403$	1.00386e6	0.307901
$404$	3.04163e6	0.927157
$405$	0	0
$406$	−671104.	−0.202057
$407$	378139.	0.113153
$408$	0	0
$409$	−4.79285e6	−1.41672	−0.708362	−	0.705849i	$-0.750566\pi$
$409$	−4.79285e6	−1.41672	−0.708362	+	0.705849i	$0.750566\pi$
$410$	1.79416e6	0.527109
$411$	0	0
$412$	2.27538e6	0.660405
$413$	779198.	0.224788
$414$	0	0
$415$	7.22540e6	2.05940
$416$	159744.	0.0452576
$417$	0	0
$418$	−538508.	−0.150748
$419$	4.84279e6	1.34760	0.673799	−	0.738915i	$-0.264661\pi$
$419$	4.84279e6	1.34760	0.673799	+	0.738915i	$0.264661\pi$
$420$	0	0
$421$	−5.84008e6	−1.60588	−0.802941	−	0.596059i	$-0.796732\pi$
$421$	−5.84008e6	−1.60588	−0.802941	+	0.596059i	$0.796732\pi$
$422$	2.14388e6	0.586029
$423$	0	0
$424$	366336.	0.0989612
$425$	3.16578e6	0.850176
$426$	0	0
$427$	−912576.	−0.242214
$428$	927552.	0.244753
$429$	0	0
$430$	−1.53681e6	−0.400819
$431$	2.40485e6	0.623584	0.311792	−	0.950150i	$-0.399071\pi$
$431$	2.40485e6	0.623584	0.311792	+	0.950150i	$0.399071\pi$
$432$	0	0
$433$	−3.08008e6	−0.789482	−0.394741	−	0.918792i	$-0.629166\pi$
$433$	−3.08008e6	−0.789482	−0.394741	+	0.918792i	$0.629166\pi$
$434$	−1.26126e6	−0.321425
$435$	0	0
$436$	−1.60290e6	−0.403821
$437$	−6.92777e6	−1.73536
$438$	0	0
$439$	−5.47711e6	−1.35641	−0.678204	−	0.734874i	$-0.737241\pi$
$439$	−5.47711e6	−1.35641	−0.678204	+	0.734874i	$0.737241\pi$
$440$	−355264.	−0.0874822
$441$	0	0
$442$	383136.	0.0932819
$443$	3.32792e6	0.805681	0.402841	−	0.915270i	$-0.368023\pi$
$443$	3.32792e6	0.805681	0.402841	+	0.915270i	$0.368023\pi$
$444$	0	0
$445$	1.11029e6	0.265789
$446$	173540.	0.0413107
$447$	0	0
$448$	−200704.	−0.0472456
$449$	−2.43221e6	−0.569358	−0.284679	−	0.958623i	$-0.591887\pi$
$449$	−2.43221e6	−0.569358	−0.284679	+	0.958623i	$0.591887\pi$
$450$	0	0
$451$	−300669.	−0.0696061
$452$	3.29642e6	0.758920
$453$	0	0
$454$	−4.85071e6	−1.10450
$455$	−695604.	−0.157519
$456$	0	0
$457$	8.05921e6	1.80510	0.902552	−	0.430581i	$-0.141691\pi$
$457$	8.05921e6	1.80510	0.902552	+	0.430581i	$0.141691\pi$
$458$	2.27531e6	0.506847
$459$	0	0
$460$	−4.57038e6	−1.00707
$461$	−1.94668e6	−0.426620	−0.213310	−	0.976985i	$-0.568424\pi$
$461$	−1.94668e6	−0.426620	−0.213310	+	0.976985i	$0.568424\pi$
$462$	0	0
$463$	−4.44786e6	−0.964270	−0.482135	−	0.876097i	$-0.660139\pi$
$463$	−4.44786e6	−0.964270	−0.482135	+	0.876097i	$0.660139\pi$
$464$	876544.	0.189007
$465$	0	0
$466$	−1.41388e6	−0.301611
$467$	5.21753e6	1.10706	0.553532	−	0.832828i	$-0.313280\pi$
$467$	5.21753e6	1.10706	0.553532	+	0.832828i	$0.313280\pi$
$468$	0	0
$469$	−582316.	−0.122244
$470$	1.87169e6	0.390831
$471$	0	0
$472$	−1.01773e6	−0.210270
$473$	257542.	0.0529292
$474$	0	0
$475$	1.13793e7	2.31410
$476$	−481376.	−0.0973794
$477$	0	0
$478$	−4.30022e6	−0.860838
$479$	1.93410e6	0.385158	0.192579	−	0.981281i	$-0.438315\pi$
$479$	1.93410e6	0.385158	0.192579	+	0.981281i	$0.438315\pi$
$480$	0	0
$481$	−967044.	−0.190583
$482$	−6.02966e6	−1.18216
$483$	0	0
$484$	−2.51728e6	−0.488448
$485$	9.47128e6	1.82833
$486$	0	0
$487$	2.82693e6	0.540122	0.270061	−	0.962843i	$-0.412956\pi$
$487$	2.82693e6	0.540122	0.270061	+	0.962843i	$0.412956\pi$
$488$	1.19194e6	0.226571
$489$	0	0
$490$	873964.	0.164438
$491$	8.52446e6	1.59574	0.797871	−	0.602828i	$-0.205959\pi$
$491$	8.52446e6	1.59574	0.797871	+	0.602828i	$0.205959\pi$
$492$	0	0
$493$	2.10234e6	0.389569
$494$	1.37717e6	0.253904
$495$	0	0
$496$	1.64736e6	0.300666
$497$	2.73807e6	0.497226
$498$	0	0
$499$	−459146.	−0.0825466	−0.0412733	−	0.999148i	$-0.513141\pi$
$499$	−459146.	−0.0825466	−0.0412733	+	0.999148i	$0.513141\pi$
$500$	2.95714e6	0.528989
$501$	0	0
$502$	200160.	0.0354501
$503$	−6.68198e6	−1.17757	−0.588783	−	0.808291i	$-0.700393\pi$
$503$	−6.68198e6	−1.17757	−0.588783	+	0.808291i	$0.700393\pi$
$504$	0	0
$505$	1.72993e7	3.01856
$506$	765916.	0.132986
$507$	0	0
$508$	3.29069e6	0.565754
$509$	−3.37000e6	−0.576548	−0.288274	−	0.957548i	$-0.593081\pi$
$509$	−3.37000e6	−0.576548	−0.288274	+	0.957548i	$0.593081\pi$
$510$	0	0
$511$	2.08221e6	0.352754
$512$	262144.	0.0441942
$513$	0	0
$514$	−3.83579e6	−0.640393
$515$	1.29412e7	2.15009
$516$	0	0
$517$	−313662.	−0.0516102
$518$	1.21500e6	0.198954
$519$	0	0
$520$	908544.	0.147346
$521$	1.36876e6	0.220919	0.110459	−	0.993881i	$-0.464768\pi$
$521$	1.36876e6	0.220919	0.110459	+	0.993881i	$0.464768\pi$
$522$	0	0
$523$	1.10535e6	0.176704	0.0883522	−	0.996089i	$-0.471840\pi$
$523$	1.10535e6	0.176704	0.0883522	+	0.996089i	$0.471840\pi$
$524$	−3.23600e6	−0.514850
$525$	0	0
$526$	6.82823e6	1.07608
$527$	3.95109e6	0.619713
$528$	0	0
$529$	3.41698e6	0.530888
$530$	2.08354e6	0.322189
$531$	0	0
$532$	−1.73029e6	−0.265057
$533$	768924.	0.117237
$534$	0	0
$535$	5.27545e6	0.796848
$536$	760576.	0.114349
$537$	0	0
$538$	1.41392e6	0.210605
$539$	−146461.	−0.0217145
$540$	0	0
$541$	3.78908e6	0.556597	0.278298	−	0.960495i	$-0.410230\pi$
$541$	3.78908e6	0.556597	0.278298	+	0.960495i	$0.410230\pi$
$542$	−9.11098e6	−1.33219
$543$	0	0
$544$	628736.	0.0910901
$545$	−9.11647e6	−1.31473
$546$	0	0
$547$	−1.38407e7	−1.97783	−0.988914	−	0.148486i	$-0.952560\pi$
$547$	−1.38407e7	−1.97783	−0.988914	+	0.148486i	$0.952560\pi$
$548$	4.52589e6	0.643802
$549$	0	0
$550$	−1.25806e6	−0.177336
$551$	7.55677e6	1.06037
$552$	0	0
$553$	1.15748e6	0.160953
$554$	−5.69504e6	−0.788355
$555$	0	0
$556$	−6.02874e6	−0.827064
$557$	1.24182e7	1.69598	0.847991	−	0.530011i	$-0.177812\pi$
$557$	1.24182e7	1.69598	0.847991	+	0.530011i	$0.177812\pi$
$558$	0	0
$559$	−658632.	−0.0891483
$560$	−1.14150e6	−0.153818
$561$	0	0
$562$	4.61295e6	0.616081
$563$	4.10598e6	0.545941	0.272970	−	0.962022i	$-0.411994\pi$
$563$	4.10598e6	0.545941	0.272970	+	0.962022i	$0.411994\pi$
$564$	0	0
$565$	1.87484e7	2.47083
$566$	−303920.	−0.0398766
$567$	0	0
$568$	−3.57626e6	−0.465112
$569$	4.34685e6	0.562852	0.281426	−	0.959583i	$-0.409193\pi$
$569$	4.34685e6	0.562852	0.281426	+	0.959583i	$0.409193\pi$
$570$	0	0
$571$	−6.57428e6	−0.843836	−0.421918	−	0.906634i	$-0.638643\pi$
$571$	−6.57428e6	−0.843836	−0.421918	+	0.906634i	$0.638643\pi$
$572$	−152256.	−0.0194574
$573$	0	0
$574$	−966084.	−0.122387
$575$	−1.61847e7	−2.04143
$576$	0	0
$577$	1.22885e7	1.53660	0.768300	−	0.640090i	$-0.221103\pi$
$577$	1.22885e7	1.53660	0.768300	+	0.640090i	$0.221103\pi$
$578$	−4.17144e6	−0.519358
$579$	0	0
$580$	4.98534e6	0.615354
$581$	−3.89060e6	−0.478163
$582$	0	0
$583$	−349164.	−0.0425459
$584$	−2.71962e6	−0.329971
$585$	0	0
$586$	−7.76993e6	−0.934702
$587$	3.47921e6	0.416759	0.208380	−	0.978048i	$-0.433181\pi$
$587$	3.47921e6	0.416759	0.208380	+	0.978048i	$0.433181\pi$
$588$	0	0
$589$	1.42020e7	1.68680
$590$	−5.78833e6	−0.684578
$591$	0	0
$592$	−1.58694e6	−0.186105
$593$	1.10998e7	1.29622	0.648109	−	0.761547i	$-0.275560\pi$
$593$	1.10998e7	1.29622	0.648109	+	0.761547i	$0.275560\pi$
$594$	0	0
$595$	−2.73783e6	−0.317040
$596$	−665184.	−0.0767054
$597$	0	0
$598$	−1.95874e6	−0.223987
$599$	−1.13402e7	−1.29138	−0.645688	−	0.763601i	$-0.723429\pi$
$599$	−1.13402e7	−1.29138	−0.645688	+	0.763601i	$0.723429\pi$
$600$	0	0
$601$	1.21989e6	0.137764	0.0688819	−	0.997625i	$-0.478057\pi$
$601$	1.21989e6	0.137764	0.0688819	+	0.997625i	$0.478057\pi$
$602$	827512.	0.0930643
$603$	0	0
$604$	−8.83491e6	−0.985394
$605$	−1.43170e7	−1.59025
$606$	0	0
$607$	9.71749e6	1.07049	0.535245	−	0.844697i	$-0.320219\pi$
$607$	9.71749e6	1.07049	0.535245	+	0.844697i	$0.320219\pi$
$608$	2.25997e6	0.247938
$609$	0	0
$610$	6.77914e6	0.737649
$611$	802152.	0.0869268
$612$	0	0
$613$	−8.77080e6	−0.942732	−0.471366	−	0.881938i	$-0.656239\pi$
$613$	−8.77080e6	−0.942732	−0.471366	+	0.881938i	$0.656239\pi$
$614$	−7.15720e6	−0.766165
$615$	0	0
$616$	191296.	0.0203121
$617$	−6.26430e6	−0.662460	−0.331230	−	0.943550i	$-0.607464\pi$
$617$	−6.26430e6	−0.662460	−0.331230	+	0.943550i	$0.607464\pi$
$618$	0	0
$619$	−1.13527e7	−1.19089	−0.595445	−	0.803396i	$-0.703024\pi$
$619$	−1.13527e7	−1.19089	−0.595445	+	0.803396i	$0.703024\pi$
$620$	9.36936e6	0.978883
$621$	0	0
$622$	−1.31631e7	−1.36422
$623$	−597849.	−0.0617123
$624$	0	0
$625$	706211.	0.0723160
$626$	5.13880e6	0.524114
$627$	0	0
$628$	−5.17907e6	−0.524026
$629$	−3.80619e6	−0.383587
$630$	0	0
$631$	1.62718e6	0.162690	0.0813452	−	0.996686i	$-0.474078\pi$
$631$	1.62718e6	0.162690	0.0813452	+	0.996686i	$0.474078\pi$
$632$	−1.51181e6	−0.150558
$633$	0	0
$634$	−7.23118e6	−0.714473
$635$	1.87158e7	1.84193
$636$	0	0
$637$	374556.	0.0365736
$638$	−835456.	−0.0812591
$639$	0	0
$640$	1.49094e6	0.143884
$641$	1.09340e7	1.05108	0.525538	−	0.850770i	$-0.323864\pi$
$641$	1.09340e7	1.05108	0.525538	+	0.850770i	$0.323864\pi$
$642$	0	0
$643$	−6.62540e6	−0.631953	−0.315977	−	0.948767i	$-0.602332\pi$
$643$	−6.62540e6	−0.631953	−0.315977	+	0.948767i	$0.602332\pi$
$644$	2.46098e6	0.233826
$645$	0	0
$646$	5.42039e6	0.511034
$647$	−1.00420e7	−0.943105	−0.471553	−	0.881838i	$-0.656306\pi$
$647$	−1.00420e7	−0.943105	−0.471553	+	0.881838i	$0.656306\pi$
$648$	0	0
$649$	970022.	0.0904003
$650$	3.21734e6	0.298685
$651$	0	0
$652$	−4.45466e6	−0.410389
$653$	1.70284e7	1.56276	0.781379	−	0.624057i	$-0.214517\pi$
$653$	1.70284e7	1.56276	0.781379	+	0.624057i	$0.214517\pi$
$654$	0	0
$655$	−1.84048e7	−1.67620
$656$	1.26182e6	0.114483
$657$	0	0
$658$	−1.00783e6	−0.0907451
$659$	−2.81993e6	−0.252944	−0.126472	−	0.991970i	$-0.540365\pi$
$659$	−2.81993e6	−0.252944	−0.126472	+	0.991970i	$0.540365\pi$
$660$	0	0
$661$	−9.37419e6	−0.834507	−0.417254	−	0.908790i	$-0.637007\pi$
$661$	−9.37419e6	−0.834507	−0.417254	+	0.908790i	$0.637007\pi$
$662$	8.72590e6	0.773865
$663$	0	0
$664$	5.08160e6	0.447281
$665$	−9.84101e6	−0.862950
$666$	0	0
$667$	−1.07479e7	−0.935428
$668$	−8.04906e6	−0.697917
$669$	0	0
$670$	4.32578e6	0.372286
$671$	−1.13606e6	−0.0974084
$672$	0	0
$673$	−1.42917e7	−1.21632	−0.608158	−	0.793816i	$-0.708091\pi$
$673$	−1.42917e7	−1.21632	−0.608158	+	0.793816i	$0.708091\pi$
$674$	−1.85317e6	−0.157132
$675$	0	0
$676$	−5.55131e6	−0.467228
$677$	−917127.	−0.0769056	−0.0384528	−	0.999260i	$-0.512243\pi$
$677$	−917127.	−0.0769056	−0.0384528	+	0.999260i	$0.512243\pi$
$678$	0	0
$679$	−5.09992e6	−0.424511
$680$	3.57594e6	0.296563
$681$	0	0
$682$	−1.57014e6	−0.129264
$683$	8.63528e6	0.708312	0.354156	−	0.935186i	$-0.384768\pi$
$683$	8.63528e6	0.708312	0.354156	+	0.935186i	$0.384768\pi$
$684$	0	0
$685$	2.57410e7	2.09604
$686$	−470596.	−0.0381802
$687$	0	0
$688$	−1.08083e6	−0.0870537
$689$	892944.	0.0716599
$690$	0	0
$691$	1.31114e7	1.04461	0.522306	−	0.852758i	$-0.325072\pi$
$691$	1.31114e7	1.04461	0.522306	+	0.852758i	$0.325072\pi$
$692$	7.52046e6	0.597007
$693$	0	0
$694$	6.52993e6	0.514648
$695$	−3.42884e7	−2.69269
$696$	0	0
$697$	3.02641e6	0.235964
$698$	1.38507e7	1.07605
$699$	0	0
$700$	−4.04230e6	−0.311806
$701$	−1.24314e7	−0.955490	−0.477745	−	0.878499i	$-0.658546\pi$
$701$	−1.24314e7	−0.955490	−0.477745	+	0.878499i	$0.658546\pi$
$702$	0	0
$703$	−1.36812e7	−1.04408
$704$	−249856.	−0.0190002
$705$	0	0
$706$	−4.29848e6	−0.324566
$707$	−9.31500e6	−0.700865
$708$	0	0
$709$	−9.35949e6	−0.699256	−0.349628	−	0.936889i	$-0.613692\pi$
$709$	−9.35949e6	−0.699256	−0.349628	+	0.936889i	$0.613692\pi$
$710$	−2.03400e7	−1.51427
$711$	0	0
$712$	780864.	0.0577265
$713$	−2.01995e7	−1.48804
$714$	0	0
$715$	−865956.	−0.0633477
$716$	−1.13353e7	−0.826324
$717$	0	0
$718$	−5.81278e6	−0.420797
$719$	−3.73119e6	−0.269169	−0.134585	−	0.990902i	$-0.542970\pi$
$719$	−3.73119e6	−0.269169	−0.134585	+	0.990902i	$0.542970\pi$
$720$	0	0
$721$	−6.96834e6	−0.499219
$722$	9.57900e6	0.683876
$723$	0	0
$724$	9.36605e6	0.664064
$725$	1.76541e7	1.24739
$726$	0	0
$727$	5.99543e6	0.420712	0.210356	−	0.977625i	$-0.432538\pi$
$727$	5.99543e6	0.420712	0.210356	+	0.977625i	$0.432538\pi$
$728$	−489216.	−0.0342115
$729$	0	0
$730$	−1.54678e7	−1.07429
$731$	−2.59231e6	−0.179429
$732$	0	0
$733$	−1.97288e6	−0.135625	−0.0678126	−	0.997698i	$-0.521602\pi$
$733$	−1.97288e6	−0.135625	−0.0678126	+	0.997698i	$0.521602\pi$
$734$	−8.16970e6	−0.559714
$735$	0	0
$736$	−3.21434e6	−0.218724
$737$	−724924.	−0.0491613
$738$	0	0
$739$	4.97822e6	0.335323	0.167661	−	0.985845i	$-0.446379\pi$
$739$	4.97822e6	0.335323	0.167661	+	0.985845i	$0.446379\pi$
$740$	−9.02574e6	−0.605904
$741$	0	0
$742$	−1.12190e6	−0.0748076
$743$	−3.88689e6	−0.258303	−0.129152	−	0.991625i	$-0.541225\pi$
$743$	−3.88689e6	−0.258303	−0.129152	+	0.991625i	$0.541225\pi$
$744$	0	0
$745$	−3.78323e6	−0.249731
$746$	1.06874e7	0.703111
$747$	0	0
$748$	−599264.	−0.0391619
$749$	−2.84063e6	−0.185016
$750$	0	0
$751$	1.67681e6	0.108489	0.0542444	−	0.998528i	$-0.482725\pi$
$751$	1.67681e6	0.108489	0.0542444	+	0.998528i	$0.482725\pi$
$752$	1.31635e6	0.0848843
$753$	0	0
$754$	2.13658e6	0.136864
$755$	−5.02486e7	−3.20816
$756$	0	0
$757$	1.75632e7	1.11395	0.556973	−	0.830530i	$-0.311963\pi$
$757$	1.75632e7	1.11395	0.556973	+	0.830530i	$0.311963\pi$
$758$	8.38098e6	0.529812
$759$	0	0
$760$	1.28536e7	0.807216
$761$	−2.18506e7	−1.36773	−0.683866	−	0.729607i	$-0.739703\pi$
$761$	−2.18506e7	−1.36773	−0.683866	+	0.729607i	$0.739703\pi$
$762$	0	0
$763$	4.90887e6	0.305260
$764$	−2.57410e6	−0.159548
$765$	0	0
$766$	2.87262e6	0.176891
$767$	−2.48071e6	−0.152261
$768$	0	0
$769$	−4.12233e6	−0.251378	−0.125689	−	0.992070i	$-0.540114\pi$
$769$	−4.12233e6	−0.251378	−0.125689	+	0.992070i	$0.540114\pi$
$770$	1.08800e6	0.0661303
$771$	0	0
$772$	−1.32608e7	−0.800806
$773$	1.84891e7	1.11293	0.556463	−	0.830872i	$-0.312158\pi$
$773$	1.84891e7	1.11293	0.556463	+	0.830872i	$0.312158\pi$
$774$	0	0
$775$	3.31789e7	1.98430
$776$	6.66112e6	0.397094
$777$	0	0
$778$	9.96328e6	0.590138
$779$	1.08783e7	0.642270
$780$	0	0
$781$	3.40862e6	0.199964
$782$	−7.70938e6	−0.450820
$783$	0	0
$784$	614656.	0.0357143
$785$	−2.94560e7	−1.70608
$786$	0	0
$787$	−1.75980e7	−1.01280	−0.506402	−	0.862298i	$-0.669025\pi$
$787$	−1.75980e7	−1.01280	−0.506402	+	0.862298i	$0.669025\pi$
$788$	8.49987e6	0.487637
$789$	0	0
$790$	−8.59841e6	−0.490174
$791$	−1.00953e7	−0.573690
$792$	0	0
$793$	2.90534e6	0.164064
$794$	−2.07430e7	−1.16767
$795$	0	0
$796$	8.44030e6	0.472145
$797$	2.67358e7	1.49090	0.745448	−	0.666564i	$-0.232236\pi$
$797$	2.67358e7	1.49090	0.745448	+	0.666564i	$0.232236\pi$
$798$	0	0
$799$	3.15719e6	0.174958
$800$	5.27974e6	0.291667
$801$	0	0
$802$	−8.21846e6	−0.451185
$803$	2.59213e6	0.141863
$804$	0	0
$805$	1.39968e7	0.761271
$806$	4.01544e6	0.217719
$807$	0	0
$808$	1.21665e7	0.655599
$809$	1.57538e7	0.846279	0.423140	−	0.906065i	$-0.360928\pi$
$809$	1.57538e7	0.846279	0.423140	+	0.906065i	$0.360928\pi$
$810$	0	0
$811$	1.93811e7	1.03473	0.517364	−	0.855765i	$-0.326913\pi$
$811$	1.93811e7	1.03473	0.517364	+	0.855765i	$0.326913\pi$
$812$	−2.68442e6	−0.142876
$813$	0	0
$814$	1.51256e6	0.0800111
$815$	−2.53359e7	−1.33611
$816$	0	0
$817$	−9.31795e6	−0.488388
$818$	−1.91714e7	−1.00178
$819$	0	0
$820$	7.17662e6	0.372722
$821$	1.75374e6	0.0908046	0.0454023	−	0.998969i	$-0.485543\pi$
$821$	1.75374e6	0.0908046	0.0454023	+	0.998969i	$0.485543\pi$
$822$	0	0
$823$	−3.28736e7	−1.69180	−0.845898	−	0.533344i	$-0.820935\pi$
$823$	−3.28736e7	−1.69180	−0.845898	+	0.533344i	$0.820935\pi$
$824$	9.10150e6	0.466977
$825$	0	0
$826$	3.11679e6	0.158949
$827$	−148647.	−0.00755775	−0.00377887	−	0.999993i	$-0.501203\pi$
$827$	−148647.	−0.00755775	−0.00377887	+	0.999993i	$0.501203\pi$
$828$	0	0
$829$	2.57770e7	1.30270	0.651352	−	0.758776i	$-0.274202\pi$
$829$	2.57770e7	1.30270	0.651352	+	0.758776i	$0.274202\pi$
$830$	2.89016e7	1.45622
$831$	0	0
$832$	638976.	0.0320019
$833$	1.47421e6	0.0736119
$834$	0	0
$835$	−4.57790e7	−2.27222
$836$	−2.15403e6	−0.106595
$837$	0	0
$838$	1.93711e7	0.952895
$839$	−2.35917e7	−1.15706	−0.578528	−	0.815662i	$-0.696373\pi$
$839$	−2.35917e7	−1.15706	−0.578528	+	0.815662i	$0.696373\pi$
$840$	0	0
$841$	−8.78737e6	−0.428419
$842$	−2.33603e7	−1.13553
$843$	0	0
$844$	8.57552e6	0.414385
$845$	−3.15731e7	−1.52116
$846$	0	0
$847$	7.70917e6	0.369232
$848$	1.46534e6	0.0699761
$849$	0	0
$850$	1.26631e7	0.601165
$851$	1.94587e7	0.921062
$852$	0	0
$853$	6.08230e6	0.286217	0.143109	−	0.989707i	$-0.454290\pi$
$853$	6.08230e6	0.286217	0.143109	+	0.989707i	$0.454290\pi$
$854$	−3.65030e6	−0.171271
$855$	0	0
$856$	3.71021e6	0.173067
$857$	1.31230e7	0.610352	0.305176	−	0.952296i	$-0.401285\pi$
$857$	1.31230e7	0.610352	0.305176	+	0.952296i	$0.401285\pi$
$858$	0	0
$859$	−1.31280e7	−0.607039	−0.303519	−	0.952825i	$-0.598162\pi$
$859$	−1.31280e7	−0.607039	−0.303519	+	0.952825i	$0.598162\pi$
$860$	−6.14723e6	−0.283422
$861$	0	0
$862$	9.61940e6	0.440941
$863$	1.43530e7	0.656020	0.328010	−	0.944674i	$-0.393622\pi$
$863$	1.43530e7	0.656020	0.328010	+	0.944674i	$0.393622\pi$
$864$	0	0
$865$	4.27726e7	1.94368
$866$	−1.23203e7	−0.558248
$867$	0	0
$868$	−5.04504e6	−0.227282
$869$	1.44094e6	0.0647287
$870$	0	0
$871$	1.85390e6	0.0828022
$872$	−6.41158e6	−0.285545
$873$	0	0
$874$	−2.77111e7	−1.22709
$875$	−9.05623e6	−0.399878
$876$	0	0
$877$	1.46782e7	0.644427	0.322214	−	0.946667i	$-0.395573\pi$
$877$	1.46782e7	0.644427	0.322214	+	0.946667i	$0.395573\pi$
$878$	−2.19084e7	−0.959125
$879$	0	0
$880$	−1.42106e6	−0.0618592
$881$	6.50392e6	0.282316	0.141158	−	0.989987i	$-0.454917\pi$
$881$	6.50392e6	0.282316	0.141158	+	0.989987i	$0.454917\pi$
$882$	0	0
$883$	−3.32985e7	−1.43722	−0.718609	−	0.695414i	$-0.755221\pi$
$883$	−3.32985e7	−1.43722	−0.718609	+	0.695414i	$0.755221\pi$
$884$	1.53254e6	0.0659602
$885$	0	0
$886$	1.33117e7	0.569703
$887$	4.26864e7	1.82171	0.910857	−	0.412722i	$-0.135422\pi$
$887$	4.26864e7	1.82171	0.910857	+	0.412722i	$0.135422\pi$
$888$	0	0
$889$	−1.00777e7	−0.427670
$890$	4.44116e6	0.187941
$891$	0	0
$892$	694160.	0.0292111
$893$	1.13484e7	0.476218
$894$	0	0
$895$	−6.44695e7	−2.69028
$896$	−802816.	−0.0334077
$897$	0	0
$898$	−9.72884e6	−0.402597
$899$	2.20334e7	0.909250
$900$	0	0
$901$	3.51454e6	0.144230
$902$	−1.20268e6	−0.0492189
$903$	0	0
$904$	1.31857e7	0.536637
$905$	5.32694e7	2.16200
$906$	0	0
$907$	2.06295e7	0.832667	0.416334	−	0.909212i	$-0.363315\pi$
$907$	2.06295e7	0.832667	0.416334	+	0.909212i	$0.363315\pi$
$908$	−1.94028e7	−0.781000
$909$	0	0
$910$	−2.78242e6	−0.111383
$911$	−3.06892e7	−1.22515	−0.612575	−	0.790412i	$-0.709866\pi$
$911$	−3.06892e7	−1.22515	−0.612575	+	0.790412i	$0.709866\pi$
$912$	0	0
$913$	−4.84340e6	−0.192297
$914$	3.22369e7	1.27640
$915$	0	0
$916$	9.10125e6	0.358395
$917$	9.91025e6	0.389190
$918$	0	0
$919$	4.94223e6	0.193034	0.0965170	−	0.995331i	$-0.469230\pi$
$919$	4.94223e6	0.193034	0.0965170	+	0.995331i	$0.469230\pi$
$920$	−1.82815e7	−0.712103
$921$	0	0
$922$	−7.78670e6	−0.301666
$923$	−8.71712e6	−0.336798
$924$	0	0
$925$	−3.19620e7	−1.22823
$926$	−1.77914e7	−0.681842
$927$	0	0
$928$	3.50618e6	0.133648
$929$	3.50874e7	1.33386	0.666932	−	0.745119i	$-0.267607\pi$
$929$	3.50874e7	1.33386	0.666932	+	0.745119i	$0.267607\pi$
$930$	0	0
$931$	5.29901e6	0.200364
$932$	−5.65552e6	−0.213271
$933$	0	0
$934$	2.08701e7	0.782812
$935$	−3.40831e6	−0.127500
$936$	0	0
$937$	−1.12272e7	−0.417757	−0.208878	−	0.977942i	$-0.566981\pi$
$937$	−1.12272e7	−0.417757	−0.208878	+	0.977942i	$0.566981\pi$
$938$	−2.32926e6	−0.0864394
$939$	0	0
$940$	7.48675e6	0.276359
$941$	−3.58550e6	−0.132000	−0.0660002	−	0.997820i	$-0.521024\pi$
$941$	−3.58550e6	−0.132000	−0.0660002	+	0.997820i	$0.521024\pi$
$942$	0	0
$943$	−1.54721e7	−0.566593
$944$	−4.07091e6	−0.148683
$945$	0	0
$946$	1.03017e6	0.0374266
$947$	−2.44504e7	−0.885954	−0.442977	−	0.896533i	$-0.646078\pi$
$947$	−2.44504e7	−0.885954	−0.442977	+	0.896533i	$0.646078\pi$
$948$	0	0
$949$	−6.62906e6	−0.238939
$950$	4.55172e7	1.63631
$951$	0	0
$952$	−1.92550e6	−0.0688576
$953$	6.77017e6	0.241472	0.120736	−	0.992685i	$-0.461475\pi$
$953$	6.77017e6	0.241472	0.120736	+	0.992685i	$0.461475\pi$
$954$	0	0
$955$	−1.46402e7	−0.519443
$956$	−1.72009e7	−0.608704
$957$	0	0
$958$	7.73638e6	0.272348
$959$	−1.38605e7	−0.486669
$960$	0	0
$961$	1.27801e7	0.446401
$962$	−3.86818e6	−0.134762
$963$	0	0
$964$	−2.41186e7	−0.835911
$965$	−7.54210e7	−2.60720
$966$	0	0
$967$	−1.96428e7	−0.675518	−0.337759	−	0.941233i	$-0.609669\pi$
$967$	−1.96428e7	−0.675518	−0.337759	+	0.941233i	$0.609669\pi$
$968$	−1.00691e7	−0.345385
$969$	0	0
$970$	3.78851e7	1.29282
$971$	4.91321e7	1.67231	0.836156	−	0.548491i	$-0.184798\pi$
$971$	4.91321e7	1.67231	0.836156	+	0.548491i	$0.184798\pi$
$972$	0	0
$973$	1.84630e7	0.625202
$974$	1.13077e7	0.381924
$975$	0	0
$976$	4.76774e6	0.160210
$977$	3.86953e7	1.29694	0.648472	−	0.761238i	$-0.275408\pi$
$977$	3.86953e7	1.29694	0.648472	+	0.761238i	$0.275408\pi$
$978$	0	0
$979$	−744261.	−0.0248181
$980$	3.49586e6	0.116276
$981$	0	0
$982$	3.40978e7	1.12836
$983$	−3.11692e7	−1.02883	−0.514413	−	0.857542i	$-0.671990\pi$
$983$	−3.11692e7	−1.02883	−0.514413	+	0.857542i	$0.671990\pi$
$984$	0	0
$985$	4.83430e7	1.58761
$986$	8.40934e6	0.275467
$987$	0	0
$988$	5.50867e6	0.179537
$989$	1.32529e7	0.430843
$990$	0	0
$991$	−1.64590e7	−0.532379	−0.266189	−	0.963921i	$-0.585765\pi$
$991$	−1.64590e7	−0.532379	−0.266189	+	0.963921i	$0.585765\pi$
$992$	6.58944e6	0.212603
$993$	0	0
$994$	1.09523e7	0.351592
$995$	4.80042e7	1.53717
$996$	0	0
$997$	5.30534e7	1.69035	0.845173	−	0.534493i	$-0.179497\pi$
$997$	5.30534e7	1.69035	0.845173	+	0.534493i	$0.179497\pi$
$998$	−1.83658e6	−0.0583693
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.6.a.d.1.1	yes	1
3.2	odd	2		378.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.6.a.a.1.1	✓	1	3.2	odd	2
378.6.a.d.1.1	yes	1	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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