LMFDB - Embedded newform 75.6.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,6,Mod(1,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, 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("75.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	75.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:	$12.0287864860$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	75.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +9.00000 q^{3} -28.0000 q^{4} +18.0000 q^{6} +132.000 q^{7} -120.000 q^{8} +81.0000 q^{9} +O(q^{10})$

\(q+2.00000 q^{2} +9.00000 q^{3} -28.0000 q^{4} +18.0000 q^{6} +132.000 q^{7} -120.000 q^{8} +81.0000 q^{9} +472.000 q^{11} -252.000 q^{12} +686.000 q^{13} +264.000 q^{14} +656.000 q^{16} +1562.00 q^{17} +162.000 q^{18} -2180.00 q^{19} +1188.00 q^{21} +944.000 q^{22} -264.000 q^{23} -1080.00 q^{24} +1372.00 q^{26} +729.000 q^{27} -3696.00 q^{28} +170.000 q^{29} +7272.00 q^{31} +5152.00 q^{32} +4248.00 q^{33} +3124.00 q^{34} -2268.00 q^{36} +142.000 q^{37} -4360.00 q^{38} +6174.00 q^{39} -16198.0 q^{41} +2376.00 q^{42} +10316.0 q^{43} -13216.0 q^{44} -528.000 q^{46} -18568.0 q^{47} +5904.00 q^{48} +617.000 q^{49} +14058.0 q^{51} -19208.0 q^{52} -21514.0 q^{53} +1458.00 q^{54} -15840.0 q^{56} -19620.0 q^{57} +340.000 q^{58} +34600.0 q^{59} -35738.0 q^{61} +14544.0 q^{62} +10692.0 q^{63} -10688.0 q^{64} +8496.00 q^{66} +5772.00 q^{67} -43736.0 q^{68} -2376.00 q^{69} -69088.0 q^{71} -9720.00 q^{72} +70526.0 q^{73} +284.000 q^{74} +61040.0 q^{76} +62304.0 q^{77} +12348.0 q^{78} +47640.0 q^{79} +6561.00 q^{81} -32396.0 q^{82} -74004.0 q^{83} -33264.0 q^{84} +20632.0 q^{86} +1530.00 q^{87} -56640.0 q^{88} -90030.0 q^{89} +90552.0 q^{91} +7392.00 q^{92} +65448.0 q^{93} -37136.0 q^{94} +46368.0 q^{96} +33502.0 q^{97} +1234.00 q^{98} +38232.0 q^{99} +O(q^{100})\)

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 $

$2$	2.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	2.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−28.0000	−0.875000
$5$	0	0
$5$	0	0
$6$	18.0000	0.204124
$7$	132.000	1.01819	0.509095	−	0.860710i	$-0.329980\pi$
$7$	132.000	1.01819	0.509095	+	0.860710i	$0.329980\pi$
$8$	−120.000	−0.662913
$9$	81.0000	0.333333
$10$	0	0
$11$	472.000	1.17614	0.588072	−	0.808809i	$-0.299887\pi$
$11$	472.000	1.17614	0.588072	+	0.808809i	$0.299887\pi$
$12$	−252.000	−0.505181
$13$	686.000	1.12581	0.562906	−	0.826521i	$-0.309683\pi$
$13$	686.000	1.12581	0.562906	+	0.826521i	$0.309683\pi$
$14$	264.000	0.359985
$15$	0	0
$16$	656.000	0.640625
$17$	1562.00	1.31087	0.655434	−	0.755253i	$-0.272486\pi$
$17$	1562.00	1.31087	0.655434	+	0.755253i	$0.272486\pi$
$18$	162.000	0.117851
$19$	−2180.00	−1.38539	−0.692696	−	0.721230i	$-0.743577\pi$
$19$	−2180.00	−1.38539	−0.692696	+	0.721230i	$0.743577\pi$
$20$	0	0
$21$	1188.00	0.587852
$22$	944.000	0.415829
$23$	−264.000	−0.104060	−0.0520301	−	0.998646i	$-0.516569\pi$
$23$	−264.000	−0.104060	−0.0520301	+	0.998646i	$0.516569\pi$
$24$	−1080.00	−0.382733
$25$	0	0
$26$	1372.00	0.398035
$27$	729.000	0.192450
$28$	−3696.00	−0.890916
$29$	170.000	0.0375365	0.0187683	−	0.999824i	$-0.494026\pi$
$29$	170.000	0.0375365	0.0187683	+	0.999824i	$0.494026\pi$
$30$	0	0
$31$	7272.00	1.35909	0.679547	−	0.733632i	$-0.262176\pi$
$31$	7272.00	1.35909	0.679547	+	0.733632i	$0.262176\pi$
$32$	5152.00	0.889408
$33$	4248.00	0.679047
$34$	3124.00	0.463462
$35$	0	0
$36$	−2268.00	−0.291667
$37$	142.000	0.0170523	0.00852617	−	0.999964i	$-0.497286\pi$
$37$	142.000	0.0170523	0.00852617	+	0.999964i	$0.497286\pi$
$38$	−4360.00	−0.489810
$39$	6174.00	0.649988
$40$	0	0
$41$	−16198.0	−1.50488	−0.752440	−	0.658661i	$-0.771123\pi$
$41$	−16198.0	−1.50488	−0.752440	+	0.658661i	$0.771123\pi$
$42$	2376.00	0.207837
$43$	10316.0	0.850825	0.425412	−	0.905000i	$-0.360129\pi$
$43$	10316.0	0.850825	0.425412	+	0.905000i	$0.360129\pi$
$44$	−13216.0	−1.02913
$45$	0	0
$46$	−528.000	−0.0367908
$47$	−18568.0	−1.22608	−0.613042	−	0.790050i	$-0.710055\pi$
$47$	−18568.0	−1.22608	−0.613042	+	0.790050i	$0.710055\pi$
$48$	5904.00	0.369865
$49$	617.000	0.0367109
$50$	0	0
$51$	14058.0	0.756830
$52$	−19208.0	−0.985085
$53$	−21514.0	−1.05204	−0.526019	−	0.850473i	$-0.676316\pi$
$53$	−21514.0	−1.05204	−0.526019	+	0.850473i	$0.676316\pi$
$54$	1458.00	0.0680414
$55$	0	0
$56$	−15840.0	−0.674971
$57$	−19620.0	−0.799856
$58$	340.000	0.0132712
$59$	34600.0	1.29404	0.647018	−	0.762475i	$-0.276016\pi$
$59$	34600.0	1.29404	0.647018	+	0.762475i	$0.276016\pi$
$60$	0	0
$61$	−35738.0	−1.22972	−0.614859	−	0.788637i	$-0.710787\pi$
$61$	−35738.0	−1.22972	−0.614859	+	0.788637i	$0.710787\pi$
$62$	14544.0	0.480512
$63$	10692.0	0.339397
$64$	−10688.0	−0.326172
$65$	0	0
$66$	8496.00	0.240079
$67$	5772.00	0.157087	0.0785433	−	0.996911i	$-0.474973\pi$
$67$	5772.00	0.157087	0.0785433	+	0.996911i	$0.474973\pi$
$68$	−43736.0	−1.14701
$69$	−2376.00	−0.0600791
$70$	0	0
$71$	−69088.0	−1.62651	−0.813255	−	0.581907i	$-0.802307\pi$
$71$	−69088.0	−1.62651	−0.813255	+	0.581907i	$0.802307\pi$
$72$	−9720.00	−0.220971
$73$	70526.0	1.54897	0.774483	−	0.632594i	$-0.218010\pi$
$73$	70526.0	1.54897	0.774483	+	0.632594i	$0.218010\pi$
$74$	284.000	0.00602891
$75$	0	0
$76$	61040.0	1.21222
$77$	62304.0	1.19754
$78$	12348.0	0.229805
$79$	47640.0	0.858824	0.429412	−	0.903109i	$-0.358721\pi$
$79$	47640.0	0.858824	0.429412	+	0.903109i	$0.358721\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	−32396.0	−0.532055
$83$	−74004.0	−1.17913	−0.589563	−	0.807723i	$-0.700700\pi$
$83$	−74004.0	−1.17913	−0.589563	+	0.807723i	$0.700700\pi$
$84$	−33264.0	−0.514371
$85$	0	0
$86$	20632.0	0.300812
$87$	1530.00	0.0216717
$88$	−56640.0	−0.779680
$89$	−90030.0	−1.20479	−0.602396	−	0.798197i	$-0.705787\pi$
$89$	−90030.0	−1.20479	−0.602396	+	0.798197i	$0.705787\pi$
$90$	0	0
$91$	90552.0	1.14629
$92$	7392.00	0.0910526
$93$	65448.0	0.784674
$94$	−37136.0	−0.433486
$95$	0	0
$96$	46368.0	0.513500
$97$	33502.0	0.361528	0.180764	−	0.983527i	$-0.442143\pi$
$97$	33502.0	0.361528	0.180764	+	0.983527i	$0.442143\pi$
$98$	1234.00	0.0129793
$99$	38232.0	0.392048
$100$	0	0
$101$	78882.0	0.769440	0.384720	−	0.923033i	$-0.374298\pi$
$101$	78882.0	0.769440	0.384720	+	0.923033i	$0.374298\pi$
$102$	28116.0	0.267580
$103$	82036.0	0.761924	0.380962	−	0.924591i	$-0.375593\pi$
$103$	82036.0	0.761924	0.380962	+	0.924591i	$0.375593\pi$
$104$	−82320.0	−0.746315
$105$	0	0
$106$	−43028.0	−0.371952
$107$	41652.0	0.351703	0.175852	−	0.984417i	$-0.443732\pi$
$107$	41652.0	0.351703	0.175852	+	0.984417i	$0.443732\pi$
$108$	−20412.0	−0.168394
$109$	104870.	0.845444	0.422722	−	0.906259i	$-0.361075\pi$
$109$	104870.	0.845444	0.422722	+	0.906259i	$0.361075\pi$
$110$	0	0
$111$	1278.00	0.00984517
$112$	86592.0	0.652278
$113$	153746.	1.13268	0.566341	−	0.824171i	$-0.308359\pi$
$113$	153746.	1.13268	0.566341	+	0.824171i	$0.308359\pi$
$114$	−39240.0	−0.282792
$115$	0	0
$116$	−4760.00	−0.0328444
$117$	55566.0	0.375271
$118$	69200.0	0.457511
$119$	206184.	1.33471
$120$	0	0
$121$	61733.0	0.383313
$122$	−71476.0	−0.434771
$123$	−145782.	−0.868843
$124$	−203616.	−1.18921
$125$	0	0
$126$	21384.0	0.119995
$127$	59372.0	0.326642	0.163321	−	0.986573i	$-0.447779\pi$
$127$	59372.0	0.326642	0.163321	+	0.986573i	$0.447779\pi$
$128$	−186240.	−1.00473
$129$	92844.0	0.491224
$130$	0	0
$131$	−98808.0	−0.503053	−0.251527	−	0.967850i	$-0.580933\pi$
$131$	−98808.0	−0.503053	−0.251527	+	0.967850i	$0.580933\pi$
$132$	−118944.	−0.594166
$133$	−287760.	−1.41059
$134$	11544.0	0.0555385
$135$	0	0
$136$	−187440.	−0.868990
$137$	−306918.	−1.39708	−0.698539	−	0.715572i	$-0.746166\pi$
$137$	−306918.	−1.39708	−0.698539	+	0.715572i	$0.746166\pi$
$138$	−4752.00	−0.0212412
$139$	−36820.0	−0.161639	−0.0808196	−	0.996729i	$-0.525754\pi$
$139$	−36820.0	−0.161639	−0.0808196	+	0.996729i	$0.525754\pi$
$140$	0	0
$141$	−167112.	−0.707880
$142$	−138176.	−0.575058
$143$	323792.	1.32412
$144$	53136.0	0.213542
$145$	0	0
$146$	141052.	0.547642
$147$	5553.00	0.0211950
$148$	−3976.00	−0.0149208
$149$	−138070.	−0.509488	−0.254744	−	0.967009i	$-0.581991\pi$
$149$	−138070.	−0.509488	−0.254744	+	0.967009i	$0.581991\pi$
$150$	0	0
$151$	−362408.	−1.29347	−0.646734	−	0.762716i	$-0.723865\pi$
$151$	−362408.	−1.29347	−0.646734	+	0.762716i	$0.723865\pi$
$152$	261600.	0.918393
$153$	126522.	0.436956
$154$	124608.	0.423393
$155$	0	0
$156$	−172872.	−0.568739
$157$	−246098.	−0.796818	−0.398409	−	0.917208i	$-0.630437\pi$
$157$	−246098.	−0.796818	−0.398409	+	0.917208i	$0.630437\pi$
$158$	95280.0	0.303640
$159$	−193626.	−0.607395
$160$	0	0
$161$	−34848.0	−0.105953
$162$	13122.0	0.0392837
$163$	−170084.	−0.501412	−0.250706	−	0.968063i	$-0.580663\pi$
$163$	−170084.	−0.501412	−0.250706	+	0.968063i	$0.580663\pi$
$164$	453544.	1.31677
$165$	0	0
$166$	−148008.	−0.416884
$167$	−274008.	−0.760277	−0.380139	−	0.924929i	$-0.624124\pi$
$167$	−274008.	−0.760277	−0.380139	+	0.924929i	$0.624124\pi$
$168$	−142560.	−0.389695
$169$	99303.0	0.267452
$170$	0	0
$171$	−176580.	−0.461797
$172$	−288848.	−0.744472
$173$	281886.	0.716075	0.358037	−	0.933707i	$-0.383446\pi$
$173$	281886.	0.716075	0.358037	+	0.933707i	$0.383446\pi$
$174$	3060.00	0.00766211
$175$	0	0
$176$	309632.	0.753467
$177$	311400.	0.747112
$178$	−180060.	−0.425958
$179$	−110240.	−0.257162	−0.128581	−	0.991699i	$-0.541042\pi$
$179$	−110240.	−0.257162	−0.128581	+	0.991699i	$0.541042\pi$
$180$	0	0
$181$	693182.	1.57272	0.786359	−	0.617770i	$-0.211964\pi$
$181$	693182.	1.57272	0.786359	+	0.617770i	$0.211964\pi$
$182$	181104.	0.405275
$183$	−321642.	−0.709978
$184$	31680.0	0.0689828
$185$	0	0
$186$	130896.	0.277424
$187$	737264.	1.54177
$188$	519904.	1.07282
$189$	96228.0	0.195951
$190$	0	0
$191$	374792.	0.743373	0.371687	−	0.928358i	$-0.378780\pi$
$191$	374792.	0.743373	0.371687	+	0.928358i	$0.378780\pi$
$192$	−96192.0	−0.188315
$193$	−247754.	−0.478771	−0.239385	−	0.970925i	$-0.576946\pi$
$193$	−247754.	−0.478771	−0.239385	+	0.970925i	$0.576946\pi$
$194$	67004.0	0.127819
$195$	0	0
$196$	−17276.0	−0.0321220
$197$	−373578.	−0.685829	−0.342914	−	0.939367i	$-0.611414\pi$
$197$	−373578.	−0.685829	−0.342914	+	0.939367i	$0.611414\pi$
$198$	76464.0	0.138610
$199$	−482840.	−0.864312	−0.432156	−	0.901799i	$-0.642247\pi$
$199$	−482840.	−0.864312	−0.432156	+	0.901799i	$0.642247\pi$
$200$	0	0
$201$	51948.0	0.0906940
$202$	157764.	0.272038
$203$	22440.0	0.0382193
$204$	−393624.	−0.662226
$205$	0	0
$206$	164072.	0.269381
$207$	−21384.0	−0.0346867
$208$	450016.	0.721223
$209$	−1.02896e6	−1.62942
$210$	0	0
$211$	−708748.	−1.09594	−0.547969	−	0.836499i	$-0.684599\pi$
$211$	−708748.	−1.09594	−0.547969	+	0.836499i	$0.684599\pi$
$212$	602392.	0.920533
$213$	−621792.	−0.939066
$214$	83304.0	0.124346
$215$	0	0
$216$	−87480.0	−0.127578
$217$	959904.	1.38382
$218$	209740.	0.298910
$219$	634734.	0.894296
$220$	0	0
$221$	1.07153e6	1.47579
$222$	2556.00	0.00348079
$223$	211036.	0.284181	0.142090	−	0.989854i	$-0.454618\pi$
$223$	211036.	0.284181	0.142090	+	0.989854i	$0.454618\pi$
$224$	680064.	0.905586
$225$	0	0
$226$	307492.	0.400463
$227$	−920828.	−1.18608	−0.593040	−	0.805173i	$-0.702072\pi$
$227$	−920828.	−1.18608	−0.593040	+	0.805173i	$0.702072\pi$
$228$	549360.	0.699874
$229$	−71130.0	−0.0896322	−0.0448161	−	0.998995i	$-0.514270\pi$
$229$	−71130.0	−0.0896322	−0.0448161	+	0.998995i	$0.514270\pi$
$230$	0	0
$231$	560736.	0.691399
$232$	−20400.0	−0.0248834
$233$	−1.00049e6	−1.20733	−0.603663	−	0.797239i	$-0.706293\pi$
$233$	−1.00049e6	−1.20733	−0.603663	+	0.797239i	$0.706293\pi$
$234$	111132.	0.132678
$235$	0	0
$236$	−968800.	−1.13228
$237$	428760.	0.495842
$238$	412368.	0.471892
$239$	−550520.	−0.623417	−0.311708	−	0.950178i	$-0.600901\pi$
$239$	−550520.	−0.623417	−0.311708	+	0.950178i	$0.600901\pi$
$240$	0	0
$241$	543682.	0.602979	0.301490	−	0.953469i	$-0.402516\pi$
$241$	543682.	0.602979	0.301490	+	0.953469i	$0.402516\pi$
$242$	123466.	0.135522
$243$	59049.0	0.0641500
$244$	1.00066e6	1.07600
$245$	0	0
$246$	−291564.	−0.307182
$247$	−1.49548e6	−1.55969
$248$	−872640.	−0.900961
$249$	−666036.	−0.680768
$250$	0	0
$251$	−659568.	−0.660808	−0.330404	−	0.943840i	$-0.607185\pi$
$251$	−659568.	−0.660808	−0.330404	+	0.943840i	$0.607185\pi$
$252$	−299376.	−0.296972
$253$	−124608.	−0.122390
$254$	118744.	0.115485
$255$	0	0
$256$	−30464.0	−0.0290527
$257$	1.54600e6	1.46008	0.730041	−	0.683403i	$-0.239501\pi$
$257$	1.54600e6	1.46008	0.730041	+	0.683403i	$0.239501\pi$
$258$	185688.	0.173674
$259$	18744.0	0.0173625
$260$	0	0
$261$	13770.0	0.0125122
$262$	−197616.	−0.177856
$263$	96856.0	0.0863450	0.0431725	−	0.999068i	$-0.486253\pi$
$263$	96856.0	0.0863450	0.0431725	+	0.999068i	$0.486253\pi$
$264$	−509760.	−0.450149
$265$	0	0
$266$	−575520.	−0.498719
$267$	−810270.	−0.695587
$268$	−161616.	−0.137451
$269$	83530.0	0.0703820	0.0351910	−	0.999381i	$-0.488796\pi$
$269$	83530.0	0.0703820	0.0351910	+	0.999381i	$0.488796\pi$
$270$	0	0
$271$	−857888.	−0.709590	−0.354795	−	0.934944i	$-0.615449\pi$
$271$	−857888.	−0.709590	−0.354795	+	0.934944i	$0.615449\pi$
$272$	1.02467e6	0.839774
$273$	814968.	0.661811
$274$	−613836.	−0.493942
$275$	0	0
$276$	66528.0	0.0525692
$277$	−647538.	−0.507068	−0.253534	−	0.967327i	$-0.581593\pi$
$277$	−647538.	−0.507068	−0.253534	+	0.967327i	$0.581593\pi$
$278$	−73640.0	−0.0571481
$279$	589032.	0.453031
$280$	0	0
$281$	−2.21480e6	−1.67328	−0.836639	−	0.547754i	$-0.815483\pi$
$281$	−2.21480e6	−1.67328	−0.836639	+	0.547754i	$0.815483\pi$
$282$	−334224.	−0.250273
$283$	156276.	0.115992	0.0579958	−	0.998317i	$-0.481529\pi$
$283$	156276.	0.115992	0.0579958	+	0.998317i	$0.481529\pi$
$284$	1.93446e6	1.42320
$285$	0	0
$286$	647584.	0.468146
$287$	−2.13814e6	−1.53225
$288$	417312.	0.296469
$289$	1.01999e6	0.718373
$290$	0	0
$291$	301518.	0.208728
$292$	−1.97473e6	−1.35535
$293$	56406.0	0.0383845	0.0191923	−	0.999816i	$-0.493891\pi$
$293$	56406.0	0.0383845	0.0191923	+	0.999816i	$0.493891\pi$
$294$	11106.0	0.00749358
$295$	0	0
$296$	−17040.0	−0.0113042
$297$	344088.	0.226349
$298$	−276140.	−0.180131
$299$	−181104.	−0.117152
$300$	0	0
$301$	1.36171e6	0.866301
$302$	−724816.	−0.457310
$303$	709938.	0.444236
$304$	−1.43008e6	−0.887516
$305$	0	0
$306$	253044.	0.154487
$307$	1.76061e6	1.06615	0.533074	−	0.846068i	$-0.321037\pi$
$307$	1.76061e6	1.06615	0.533074	+	0.846068i	$0.321037\pi$
$308$	−1.74451e6	−1.04785
$309$	738324.	0.439897
$310$	0	0
$311$	−1.71389e6	−1.00480	−0.502402	−	0.864634i	$-0.667550\pi$
$311$	−1.71389e6	−1.00480	−0.502402	+	0.864634i	$0.667550\pi$
$312$	−740880.	−0.430885
$313$	1.42877e6	0.824328	0.412164	−	0.911110i	$-0.364773\pi$
$313$	1.42877e6	0.824328	0.412164	+	0.911110i	$0.364773\pi$
$314$	−492196.	−0.281718
$315$	0	0
$316$	−1.33392e6	−0.751471
$317$	744382.	0.416052	0.208026	−	0.978123i	$-0.433296\pi$
$317$	744382.	0.416052	0.208026	+	0.978123i	$0.433296\pi$
$318$	−387252.	−0.214746
$319$	80240.0	0.0441483
$320$	0	0
$321$	374868.	0.203056
$322$	−69696.0	−0.0374600
$323$	−3.40516e6	−1.81606
$324$	−183708.	−0.0972222
$325$	0	0
$326$	−340168.	−0.177276
$327$	943830.	0.488118
$328$	1.94376e6	0.997604
$329$	−2.45098e6	−1.24839
$330$	0	0
$331$	−136908.	−0.0686845	−0.0343423	−	0.999410i	$-0.510934\pi$
$331$	−136908.	−0.0686845	−0.0343423	+	0.999410i	$0.510934\pi$
$332$	2.07211e6	1.03173
$333$	11502.0	0.00568411
$334$	−548016.	−0.268799
$335$	0	0
$336$	779328.	0.376593
$337$	2.43594e6	1.16840	0.584201	−	0.811609i	$-0.301408\pi$
$337$	2.43594e6	1.16840	0.584201	+	0.811609i	$0.301408\pi$
$338$	198606.	0.0945585
$339$	1.38371e6	0.653954
$340$	0	0
$341$	3.43238e6	1.59849
$342$	−353160.	−0.163270
$343$	−2.13708e6	−0.980811
$344$	−1.23792e6	−0.564023
$345$	0	0
$346$	563772.	0.253171
$347$	−1.16127e6	−0.517736	−0.258868	−	0.965913i	$-0.583350\pi$
$347$	−1.16127e6	−0.517736	−0.258868	+	0.965913i	$0.583350\pi$
$348$	−42840.0	−0.0189628
$349$	725830.	0.318986	0.159493	−	0.987199i	$-0.449014\pi$
$349$	725830.	0.318986	0.159493	+	0.987199i	$0.449014\pi$
$350$	0	0
$351$	500094.	0.216663
$352$	2.43174e6	1.04607
$353$	3.94867e6	1.68661	0.843303	−	0.537438i	$-0.180608\pi$
$353$	3.94867e6	1.68661	0.843303	+	0.537438i	$0.180608\pi$
$354$	622800.	0.264144
$355$	0	0
$356$	2.52084e6	1.05419
$357$	1.85566e6	0.770596
$358$	−220480.	−0.0909205
$359$	1.21188e6	0.496276	0.248138	−	0.968725i	$-0.420181\pi$
$359$	1.21188e6	0.496276	0.248138	+	0.968725i	$0.420181\pi$
$360$	0	0
$361$	2.27630e6	0.919309
$362$	1.38636e6	0.556040
$363$	555597.	0.221306
$364$	−2.53546e6	−1.00300
$365$	0	0
$366$	−643284.	−0.251015
$367$	−3.92999e6	−1.52309	−0.761546	−	0.648111i	$-0.775559\pi$
$367$	−3.92999e6	−1.52309	−0.761546	+	0.648111i	$0.775559\pi$
$368$	−173184.	−0.0666635
$369$	−1.31204e6	−0.501627
$370$	0	0
$371$	−2.83985e6	−1.07117
$372$	−1.83254e6	−0.686589
$373$	−1.08519e6	−0.403864	−0.201932	−	0.979400i	$-0.564722\pi$
$373$	−1.08519e6	−0.403864	−0.201932	+	0.979400i	$0.564722\pi$
$374$	1.47453e6	0.545097
$375$	0	0
$376$	2.22816e6	0.812787
$377$	116620.	0.0422590
$378$	192456.	0.0692791
$379$	4.06654e6	1.45421	0.727105	−	0.686526i	$-0.240865\pi$
$379$	4.06654e6	1.45421	0.727105	+	0.686526i	$0.240865\pi$
$380$	0	0
$381$	534348.	0.188587
$382$	749584.	0.262822
$383$	1.53454e6	0.534540	0.267270	−	0.963622i	$-0.413878\pi$
$383$	1.53454e6	0.534540	0.267270	+	0.963622i	$0.413878\pi$
$384$	−1.67616e6	−0.580079
$385$	0	0
$386$	−495508.	−0.169271
$387$	835596.	0.283608
$388$	−938056.	−0.316337
$389$	1.90737e6	0.639088	0.319544	−	0.947571i	$-0.396470\pi$
$389$	1.90737e6	0.639088	0.319544	+	0.947571i	$0.396470\pi$
$390$	0	0
$391$	−412368.	−0.136409
$392$	−74040.0	−0.0243361
$393$	−889272.	−0.290438
$394$	−747156.	−0.242477
$395$	0	0
$396$	−1.07050e6	−0.343042
$397$	1.71162e6	0.545044	0.272522	−	0.962150i	$-0.412142\pi$
$397$	1.71162e6	0.545044	0.272522	+	0.962150i	$0.412142\pi$
$398$	−965680.	−0.305580
$399$	−2.58984e6	−0.814405
$400$	0	0
$401$	2.46268e6	0.764799	0.382400	−	0.923997i	$-0.375098\pi$
$401$	2.46268e6	0.764799	0.382400	+	0.923997i	$0.375098\pi$
$402$	103896.	0.0320652
$403$	4.98859e6	1.53008
$404$	−2.20870e6	−0.673260
$405$	0	0
$406$	44880.0	0.0135126
$407$	67024.0	0.0200560
$408$	−1.68696e6	−0.501712
$409$	−3.69703e6	−1.09281	−0.546405	−	0.837521i	$-0.684004\pi$
$409$	−3.69703e6	−1.09281	−0.546405	+	0.837521i	$0.684004\pi$
$410$	0	0
$411$	−2.76226e6	−0.806604
$412$	−2.29701e6	−0.666683
$413$	4.56720e6	1.31757
$414$	−42768.0	−0.0122636
$415$	0	0
$416$	3.53427e6	1.00131
$417$	−331380.	−0.0933225
$418$	−2.05792e6	−0.576087
$419$	−1.93060e6	−0.537226	−0.268613	−	0.963248i	$-0.586565\pi$
$419$	−1.93060e6	−0.537226	−0.268613	+	0.963248i	$0.586565\pi$
$420$	0	0
$421$	−4.83910e6	−1.33064	−0.665318	−	0.746560i	$-0.731704\pi$
$421$	−4.83910e6	−1.33064	−0.665318	+	0.746560i	$0.731704\pi$
$422$	−1.41750e6	−0.387472
$423$	−1.50401e6	−0.408695
$424$	2.58168e6	0.697409
$425$	0	0
$426$	−1.24358e6	−0.332010
$427$	−4.71742e6	−1.25209
$428$	−1.16626e6	−0.307740
$429$	2.91413e6	0.764479
$430$	0	0
$431$	−2.74325e6	−0.711331	−0.355666	−	0.934613i	$-0.615746\pi$
$431$	−2.74325e6	−0.711331	−0.355666	+	0.934613i	$0.615746\pi$
$432$	478224.	0.123288
$433$	−6.05823e6	−1.55284	−0.776419	−	0.630217i	$-0.782966\pi$
$433$	−6.05823e6	−1.55284	−0.776419	+	0.630217i	$0.782966\pi$
$434$	1.91981e6	0.489253
$435$	0	0
$436$	−2.93636e6	−0.739764
$437$	575520.	0.144164
$438$	1.26947e6	0.316182
$439$	5.21044e6	1.29037	0.645183	−	0.764028i	$-0.276781\pi$
$439$	5.21044e6	1.29037	0.645183	+	0.764028i	$0.276781\pi$
$440$	0	0
$441$	49977.0	0.0122370
$442$	2.14306e6	0.521770
$443$	−1.17248e6	−0.283856	−0.141928	−	0.989877i	$-0.545330\pi$
$443$	−1.17248e6	−0.283856	−0.141928	+	0.989877i	$0.545330\pi$
$444$	−35784.0	−0.00861453
$445$	0	0
$446$	422072.	0.100473
$447$	−1.24263e6	−0.294153
$448$	−1.41082e6	−0.332105
$449$	4.14361e6	0.969981	0.484990	−	0.874520i	$-0.338823\pi$
$449$	4.14361e6	0.969981	0.484990	+	0.874520i	$0.338823\pi$
$450$	0	0
$451$	−7.64546e6	−1.76995
$452$	−4.30489e6	−0.991096
$453$	−3.26167e6	−0.746784
$454$	−1.84166e6	−0.419343
$455$	0	0
$456$	2.35440e6	0.530235
$457$	5.41338e6	1.21249	0.606245	−	0.795278i	$-0.292675\pi$
$457$	5.41338e6	1.21249	0.606245	+	0.795278i	$0.292675\pi$
$458$	−142260.	−0.0316898
$459$	1.13870e6	0.252277
$460$	0	0
$461$	5.36680e6	1.17615	0.588076	−	0.808806i	$-0.299886\pi$
$461$	5.36680e6	1.17615	0.588076	+	0.808806i	$0.299886\pi$
$462$	1.12147e6	0.244446
$463$	5.33284e6	1.15613	0.578064	−	0.815992i	$-0.303808\pi$
$463$	5.33284e6	1.15613	0.578064	+	0.815992i	$0.303808\pi$
$464$	111520.	0.0240468
$465$	0	0
$466$	−2.00099e6	−0.426854
$467$	−5.56831e6	−1.18149	−0.590746	−	0.806857i	$-0.701167\pi$
$467$	−5.56831e6	−1.18149	−0.590746	+	0.806857i	$0.701167\pi$
$468$	−1.55585e6	−0.328362
$469$	761904.	0.159944
$470$	0	0
$471$	−2.21488e6	−0.460043
$472$	−4.15200e6	−0.857832
$473$	4.86915e6	1.00069
$474$	857520.	0.175307
$475$	0	0
$476$	−5.77315e6	−1.16787
$477$	−1.74263e6	−0.350679
$478$	−1.10104e6	−0.220411
$479$	−1.57092e6	−0.312835	−0.156417	−	0.987691i	$-0.549995\pi$
$479$	−1.57092e6	−0.312835	−0.156417	+	0.987691i	$0.549995\pi$
$480$	0	0
$481$	97412.0	0.0191977
$482$	1.08736e6	0.213185
$483$	−313632.	−0.0611720
$484$	−1.72852e6	−0.335399
$485$	0	0
$486$	118098.	0.0226805
$487$	7.74497e6	1.47978	0.739891	−	0.672727i	$-0.234877\pi$
$487$	7.74497e6	1.47978	0.739891	+	0.672727i	$0.234877\pi$
$488$	4.28856e6	0.815196
$489$	−1.53076e6	−0.289490
$490$	0	0
$491$	1.97715e6	0.370115	0.185057	−	0.982728i	$-0.440753\pi$
$491$	1.97715e6	0.370115	0.185057	+	0.982728i	$0.440753\pi$
$492$	4.08190e6	0.760237
$493$	265540.	0.0492054
$494$	−2.99096e6	−0.551434
$495$	0	0
$496$	4.77043e6	0.870670
$497$	−9.11962e6	−1.65610
$498$	−1.33207e6	−0.240688
$499$	−6.55994e6	−1.17937	−0.589683	−	0.807635i	$-0.700747\pi$
$499$	−6.55994e6	−1.17937	−0.589683	+	0.807635i	$0.700747\pi$
$500$	0	0
$501$	−2.46607e6	−0.438946
$502$	−1.31914e6	−0.233631
$503$	5.92850e6	1.04478	0.522390	−	0.852707i	$-0.325041\pi$
$503$	5.92850e6	1.04478	0.522390	+	0.852707i	$0.325041\pi$
$504$	−1.28304e6	−0.224990
$505$	0	0
$506$	−249216.	−0.0432713
$507$	893727.	0.154413
$508$	−1.66242e6	−0.285812
$509$	−1.03055e6	−0.176309	−0.0881545	−	0.996107i	$-0.528097\pi$
$509$	−1.03055e6	−0.176309	−0.0881545	+	0.996107i	$0.528097\pi$
$510$	0	0
$511$	9.30943e6	1.57714
$512$	5.89875e6	0.994455
$513$	−1.58922e6	−0.266619
$514$	3.09200e6	0.516217
$515$	0	0
$516$	−2.59963e6	−0.429821
$517$	−8.76410e6	−1.44205
$518$	37488.0	0.00613858
$519$	2.53697e6	0.413426
$520$	0	0
$521$	8.48280e6	1.36913	0.684566	−	0.728951i	$-0.259992\pi$
$521$	8.48280e6	1.36913	0.684566	+	0.728951i	$0.259992\pi$
$522$	27540.0	0.00442372
$523$	−8.93676e6	−1.42865	−0.714325	−	0.699814i	$-0.753266\pi$
$523$	−8.93676e6	−1.42865	−0.714325	+	0.699814i	$0.753266\pi$
$524$	2.76662e6	0.440172
$525$	0	0
$526$	193712.	0.0305276
$527$	1.13589e7	1.78159
$528$	2.78669e6	0.435014
$529$	−6.36665e6	−0.989171
$530$	0	0
$531$	2.80260e6	0.431345
$532$	8.05728e6	1.23427
$533$	−1.11118e7	−1.69421
$534$	−1.62054e6	−0.245927
$535$	0	0
$536$	−692640.	−0.104135
$537$	−992160.	−0.148472
$538$	167060.	0.0248838
$539$	291224.	0.0431773
$540$	0	0
$541$	4.14394e6	0.608724	0.304362	−	0.952556i	$-0.401557\pi$
$541$	4.14394e6	0.608724	0.304362	+	0.952556i	$0.401557\pi$
$542$	−1.71578e6	−0.250878
$543$	6.23864e6	0.908009
$544$	8.04742e6	1.16590
$545$	0	0
$546$	1.62994e6	0.233986
$547$	−2.63115e6	−0.375991	−0.187995	−	0.982170i	$-0.560199\pi$
$547$	−2.63115e6	−0.375991	−0.187995	+	0.982170i	$0.560199\pi$
$548$	8.59370e6	1.22244
$549$	−2.89478e6	−0.409906
$550$	0	0
$551$	−370600.	−0.0520028
$552$	285120.	0.0398272
$553$	6.28848e6	0.874446
$554$	−1.29508e6	−0.179275
$555$	0	0
$556$	1.03096e6	0.141434
$557$	1.24917e7	1.70602	0.853012	−	0.521892i	$-0.174773\pi$
$557$	1.24917e7	1.70602	0.853012	+	0.521892i	$0.174773\pi$
$558$	1.17806e6	0.160171
$559$	7.07678e6	0.957869
$560$	0	0
$561$	6.63538e6	0.890140
$562$	−4.42960e6	−0.591593
$563$	−7.63892e6	−1.01569	−0.507845	−	0.861448i	$-0.669558\pi$
$563$	−7.63892e6	−1.01569	−0.507845	+	0.861448i	$0.669558\pi$
$564$	4.67914e6	0.619395
$565$	0	0
$566$	312552.	0.0410092
$567$	866052.	0.113132
$568$	8.29056e6	1.07823
$569$	455250.	0.0589480	0.0294740	−	0.999566i	$-0.490617\pi$
$569$	455250.	0.0589480	0.0294740	+	0.999566i	$0.490617\pi$
$570$	0	0
$571$	−259228.	−0.0332730	−0.0166365	−	0.999862i	$-0.505296\pi$
$571$	−259228.	−0.0332730	−0.0166365	+	0.999862i	$0.505296\pi$
$572$	−9.06618e6	−1.15860
$573$	3.37313e6	0.429187
$574$	−4.27627e6	−0.541733
$575$	0	0
$576$	−865728.	−0.108724
$577$	1.33636e7	1.67103	0.835513	−	0.549470i	$-0.185170\pi$
$577$	1.33636e7	1.67103	0.835513	+	0.549470i	$0.185170\pi$
$578$	2.03997e6	0.253983
$579$	−2.22979e6	−0.276418
$580$	0	0
$581$	−9.76853e6	−1.20057
$582$	603036.	0.0737965
$583$	−1.01546e7	−1.23735
$584$	−8.46312e6	−1.02683
$585$	0	0
$586$	112812.	0.0135710
$587$	1.22332e7	1.46536	0.732679	−	0.680574i	$-0.238270\pi$
$587$	1.22332e7	1.46536	0.732679	+	0.680574i	$0.238270\pi$
$588$	−155484.	−0.0185457
$589$	−1.58530e7	−1.88288
$590$	0	0
$591$	−3.36220e6	−0.395963
$592$	93152.0	0.0109242
$593$	−1.17588e7	−1.37318	−0.686589	−	0.727046i	$-0.740893\pi$
$593$	−1.17588e7	−1.37318	−0.686589	+	0.727046i	$0.740893\pi$
$594$	688176.	0.0800264
$595$	0	0
$596$	3.86596e6	0.445802
$597$	−4.34556e6	−0.499011
$598$	−362208.	−0.0414195
$599$	8.58404e6	0.977518	0.488759	−	0.872419i	$-0.337450\pi$
$599$	8.58404e6	0.977518	0.488759	+	0.872419i	$0.337450\pi$
$600$	0	0
$601$	−1.59372e6	−0.179980	−0.0899902	−	0.995943i	$-0.528684\pi$
$601$	−1.59372e6	−0.179980	−0.0899902	+	0.995943i	$0.528684\pi$
$602$	2.72342e6	0.306284
$603$	467532.	0.0523622
$604$	1.01474e7	1.13178
$605$	0	0
$606$	1.41988e6	0.157061
$607$	−1.52801e7	−1.68327	−0.841637	−	0.540044i	$-0.818408\pi$
$607$	−1.52801e7	−1.68327	−0.841637	+	0.540044i	$0.818408\pi$
$608$	−1.12314e7	−1.23218
$609$	201960.	0.0220659
$610$	0	0
$611$	−1.27376e7	−1.38034
$612$	−3.54262e6	−0.382336
$613$	1.67004e7	1.79504	0.897521	−	0.440971i	$-0.145366\pi$
$613$	1.67004e7	1.79504	0.897521	+	0.440971i	$0.145366\pi$
$614$	3.52122e6	0.376940
$615$	0	0
$616$	−7.47648e6	−0.793863
$617$	−1.03986e7	−1.09966	−0.549832	−	0.835275i	$-0.685308\pi$
$617$	−1.03986e7	−1.09966	−0.549832	+	0.835275i	$0.685308\pi$
$618$	1.47665e6	0.155527
$619$	−1.22631e7	−1.28640	−0.643199	−	0.765699i	$-0.722393\pi$
$619$	−1.22631e7	−1.28640	−0.643199	+	0.765699i	$0.722393\pi$
$620$	0	0
$621$	−192456.	−0.0200264
$622$	−3.42778e6	−0.355252
$623$	−1.18840e7	−1.22671
$624$	4.05014e6	0.416398
$625$	0	0
$626$	2.85753e6	0.291444
$627$	−9.26064e6	−0.940746
$628$	6.89074e6	0.697215
$629$	221804.	0.0223534
$630$	0	0
$631$	1.22509e7	1.22488	0.612442	−	0.790515i	$-0.290187\pi$
$631$	1.22509e7	1.22488	0.612442	+	0.790515i	$0.290187\pi$
$632$	−5.71680e6	−0.569325
$633$	−6.37873e6	−0.632740
$634$	1.48876e6	0.147097
$635$	0	0
$636$	5.42153e6	0.531470
$637$	423262.	0.0413296
$638$	160480.	0.0156088
$639$	−5.59613e6	−0.542170
$640$	0	0
$641$	−6.85904e6	−0.659353	−0.329677	−	0.944094i	$-0.606940\pi$
$641$	−6.85904e6	−0.659353	−0.329677	+	0.944094i	$0.606940\pi$
$642$	749736.	0.0717911
$643$	3.31120e6	0.315833	0.157916	−	0.987452i	$-0.449522\pi$
$643$	3.31120e6	0.315833	0.157916	+	0.987452i	$0.449522\pi$
$644$	975744.	0.0927089
$645$	0	0
$646$	−6.81032e6	−0.642076
$647$	−9.91821e6	−0.931478	−0.465739	−	0.884922i	$-0.654211\pi$
$647$	−9.91821e6	−0.931478	−0.465739	+	0.884922i	$0.654211\pi$
$648$	−787320.	−0.0736570
$649$	1.63312e7	1.52197
$650$	0	0
$651$	8.63914e6	0.798947
$652$	4.76235e6	0.438735
$653$	1.39102e7	1.27659	0.638294	−	0.769793i	$-0.279641\pi$
$653$	1.39102e7	1.27659	0.638294	+	0.769793i	$0.279641\pi$
$654$	1.88766e6	0.172576
$655$	0	0
$656$	−1.06259e7	−0.964063
$657$	5.71261e6	0.516322
$658$	−4.90195e6	−0.441372
$659$	−3.27884e6	−0.294108	−0.147054	−	0.989128i	$-0.546979\pi$
$659$	−3.27884e6	−0.294108	−0.147054	+	0.989128i	$0.546979\pi$
$660$	0	0
$661$	−7.00330e6	−0.623446	−0.311723	−	0.950173i	$-0.600906\pi$
$661$	−7.00330e6	−0.623446	−0.311723	+	0.950173i	$0.600906\pi$
$662$	−273816.	−0.0242836
$663$	9.64379e6	0.852048
$664$	8.88048e6	0.781657
$665$	0	0
$666$	23004.0	0.00200964
$667$	−44880.0	−0.00390605
$668$	7.67222e6	0.665243
$669$	1.89932e6	0.164072
$670$	0	0
$671$	−1.68683e7	−1.44633
$672$	6.12058e6	0.522840
$673$	−1.09747e7	−0.934019	−0.467009	−	0.884252i	$-0.654669\pi$
$673$	−1.09747e7	−0.934019	−0.467009	+	0.884252i	$0.654669\pi$
$674$	4.87188e6	0.413092
$675$	0	0
$676$	−2.78048e6	−0.234020
$677$	8.90482e6	0.746713	0.373356	−	0.927688i	$-0.378207\pi$
$677$	8.90482e6	0.746713	0.373356	+	0.927688i	$0.378207\pi$
$678$	2.76743e6	0.231208
$679$	4.42226e6	0.368104
$680$	0	0
$681$	−8.28745e6	−0.684783
$682$	6.86477e6	0.565152
$683$	−1.61956e7	−1.32845	−0.664227	−	0.747531i	$-0.731239\pi$
$683$	−1.61956e7	−1.32845	−0.664227	+	0.747531i	$0.731239\pi$
$684$	4.94424e6	0.404072
$685$	0	0
$686$	−4.27416e6	−0.346769
$687$	−640170.	−0.0517492
$688$	6.76730e6	0.545060
$689$	−1.47586e7	−1.18440
$690$	0	0
$691$	−1.51015e6	−0.120316	−0.0601581	−	0.998189i	$-0.519160\pi$
$691$	−1.51015e6	−0.120316	−0.0601581	+	0.998189i	$0.519160\pi$
$692$	−7.89281e6	−0.626565
$693$	5.04662e6	0.399179
$694$	−2.32254e6	−0.183047
$695$	0	0
$696$	−183600.	−0.0143665
$697$	−2.53013e7	−1.97270
$698$	1.45166e6	0.112779
$699$	−9.00445e6	−0.697050
$700$	0	0
$701$	1.80508e7	1.38740	0.693698	−	0.720266i	$-0.255980\pi$
$701$	1.80508e7	1.38740	0.693698	+	0.720266i	$0.255980\pi$
$702$	1.00019e6	0.0766018
$703$	−309560.	−0.0236242
$704$	−5.04474e6	−0.383625
$705$	0	0
$706$	7.89733e6	0.596305
$707$	1.04124e7	0.783436
$708$	−8.71920e6	−0.653723
$709$	1.57918e7	1.17982	0.589912	−	0.807468i	$-0.299163\pi$
$709$	1.57918e7	1.17982	0.589912	+	0.807468i	$0.299163\pi$
$710$	0	0
$711$	3.85884e6	0.286275
$712$	1.08036e7	0.798672
$713$	−1.91981e6	−0.141428
$714$	3.71131e6	0.272447
$715$	0	0
$716$	3.08672e6	0.225017
$717$	−4.95468e6	−0.359930
$718$	2.42376e6	0.175460
$719$	897600.	0.0647531	0.0323766	−	0.999476i	$-0.489692\pi$
$719$	897600.	0.0647531	0.0323766	+	0.999476i	$0.489692\pi$
$720$	0	0
$721$	1.08288e7	0.775783
$722$	4.55260e6	0.325025
$723$	4.89314e6	0.348130
$724$	−1.94091e7	−1.37613
$725$	0	0
$726$	1.11119e6	0.0782435
$727$	−8.33311e6	−0.584751	−0.292376	−	0.956304i	$-0.594446\pi$
$727$	−8.33311e6	−0.584751	−0.292376	+	0.956304i	$0.594446\pi$
$728$	−1.08662e7	−0.759890
$729$	531441.	0.0370370
$730$	0	0
$731$	1.61136e7	1.11532
$732$	9.00598e6	0.621231
$733$	6.05633e6	0.416341	0.208171	−	0.978093i	$-0.433249\pi$
$733$	6.05633e6	0.416341	0.208171	+	0.978093i	$0.433249\pi$
$734$	−7.85998e6	−0.538494
$735$	0	0
$736$	−1.36013e6	−0.0925519
$737$	2.72438e6	0.184756
$738$	−2.62408e6	−0.177352
$739$	5.19646e6	0.350023	0.175011	−	0.984566i	$-0.444004\pi$
$739$	5.19646e6	0.350023	0.175011	+	0.984566i	$0.444004\pi$
$740$	0	0
$741$	−1.34593e7	−0.900487
$742$	−5.67970e6	−0.378717
$743$	−2.67222e6	−0.177583	−0.0887914	−	0.996050i	$-0.528300\pi$
$743$	−2.67222e6	−0.177583	−0.0887914	+	0.996050i	$0.528300\pi$
$744$	−7.85376e6	−0.520170
$745$	0	0
$746$	−2.17039e6	−0.142788
$747$	−5.99432e6	−0.393042
$748$	−2.06434e7	−1.34905
$749$	5.49806e6	0.358101
$750$	0	0
$751$	1.49981e7	0.970366	0.485183	−	0.874413i	$-0.338753\pi$
$751$	1.49981e7	0.970366	0.485183	+	0.874413i	$0.338753\pi$
$752$	−1.21806e7	−0.785461
$753$	−5.93611e6	−0.381518
$754$	233240.	0.0149408
$755$	0	0
$756$	−2.69438e6	−0.171457
$757$	2.56846e7	1.62905	0.814523	−	0.580131i	$-0.196999\pi$
$757$	2.56846e7	1.62905	0.814523	+	0.580131i	$0.196999\pi$
$758$	8.13308e6	0.514141
$759$	−1.12147e6	−0.0706617
$760$	0	0
$761$	6.07960e6	0.380552	0.190276	−	0.981731i	$-0.439062\pi$
$761$	6.07960e6	0.380552	0.190276	+	0.981731i	$0.439062\pi$
$762$	1.06870e6	0.0666756
$763$	1.38428e7	0.860823
$764$	−1.04942e7	−0.650452
$765$	0	0
$766$	3.06907e6	0.188988
$767$	2.37356e7	1.45684
$768$	−274176.	−0.0167736
$769$	−2.82501e7	−1.72268	−0.861339	−	0.508030i	$-0.830374\pi$
$769$	−2.82501e7	−1.72268	−0.861339	+	0.508030i	$0.830374\pi$
$770$	0	0
$771$	1.39140e7	0.842979
$772$	6.93711e6	0.418924
$773$	2.11430e7	1.27268	0.636339	−	0.771409i	$-0.280448\pi$
$773$	2.11430e7	1.27268	0.636339	+	0.771409i	$0.280448\pi$
$774$	1.67119e6	0.100271
$775$	0	0
$776$	−4.02024e6	−0.239661
$777$	168696.	0.0100243
$778$	3.81474e6	0.225952
$779$	3.53116e7	2.08485
$780$	0	0
$781$	−3.26095e7	−1.91301
$782$	−824736.	−0.0482279
$783$	123930.	0.00722391
$784$	404752.	0.0235179
$785$	0	0
$786$	−1.77854e6	−0.102685
$787$	2.42307e7	1.39454	0.697268	−	0.716811i	$-0.254399\pi$
$787$	2.42307e7	1.39454	0.697268	+	0.716811i	$0.254399\pi$
$788$	1.04602e7	0.600100
$789$	871704.	0.0498513
$790$	0	0
$791$	2.02945e7	1.15328
$792$	−4.58784e6	−0.259893
$793$	−2.45163e7	−1.38443
$794$	3.42324e6	0.192702
$795$	0	0
$796$	1.35195e7	0.756273
$797$	−1.44322e6	−0.0804797	−0.0402398	−	0.999190i	$-0.512812\pi$
$797$	−1.44322e6	−0.0804797	−0.0402398	+	0.999190i	$0.512812\pi$
$798$	−5.17968e6	−0.287936
$799$	−2.90032e7	−1.60723
$800$	0	0
$801$	−7.29243e6	−0.401597
$802$	4.92536e6	0.270397
$803$	3.32883e7	1.82181
$804$	−1.45454e6	−0.0793573
$805$	0	0
$806$	9.97718e6	0.540967
$807$	751770.	0.0406351
$808$	−9.46584e6	−0.510071
$809$	2.99186e7	1.60720	0.803599	−	0.595171i	$-0.202916\pi$
$809$	2.99186e7	1.60720	0.803599	+	0.595171i	$0.202916\pi$
$810$	0	0
$811$	−5.30183e6	−0.283057	−0.141528	−	0.989934i	$-0.545202\pi$
$811$	−5.30183e6	−0.283057	−0.141528	+	0.989934i	$0.545202\pi$
$812$	−628320.	−0.0334419
$813$	−7.72099e6	−0.409682
$814$	134048.	0.00709087
$815$	0	0
$816$	9.22205e6	0.484844
$817$	−2.24889e7	−1.17873
$818$	−7.39406e6	−0.386367
$819$	7.33471e6	0.382097
$820$	0	0
$821$	4.55008e6	0.235592	0.117796	−	0.993038i	$-0.462417\pi$
$821$	4.55008e6	0.235592	0.117796	+	0.993038i	$0.462417\pi$
$822$	−5.52452e6	−0.285178
$823$	2.42775e7	1.24941	0.624705	−	0.780861i	$-0.285219\pi$
$823$	2.42775e7	1.24941	0.624705	+	0.780861i	$0.285219\pi$
$824$	−9.84432e6	−0.505089
$825$	0	0
$826$	9.13440e6	0.465833
$827$	−3.63999e6	−0.185070	−0.0925350	−	0.995709i	$-0.529497\pi$
$827$	−3.63999e6	−0.185070	−0.0925350	+	0.995709i	$0.529497\pi$
$828$	598752.	0.0303509
$829$	1.98674e7	1.00405	0.502023	−	0.864854i	$-0.332589\pi$
$829$	1.98674e7	1.00405	0.502023	+	0.864854i	$0.332589\pi$
$830$	0	0
$831$	−5.82784e6	−0.292756
$832$	−7.33197e6	−0.367208
$833$	963754.	0.0481231
$834$	−662760.	−0.0329945
$835$	0	0
$836$	2.88109e7	1.42574
$837$	5.30129e6	0.261558
$838$	−3.86120e6	−0.189938
$839$	−3.58666e7	−1.75908	−0.879539	−	0.475826i	$-0.842149\pi$
$839$	−3.58666e7	−1.75908	−0.879539	+	0.475826i	$0.842149\pi$
$840$	0	0
$841$	−2.04822e7	−0.998591
$842$	−9.67820e6	−0.470451
$843$	−1.99332e7	−0.966068
$844$	1.98449e7	0.958945
$845$	0	0
$846$	−3.00802e6	−0.144495
$847$	8.14876e6	0.390286
$848$	−1.41132e7	−0.673962
$849$	1.40648e6	0.0669677
$850$	0	0
$851$	−37488.0	−0.00177447
$852$	1.74102e7	0.821683
$853$	−2.25791e7	−1.06251	−0.531256	−	0.847212i	$-0.678280\pi$
$853$	−2.25791e7	−1.06251	−0.531256	+	0.847212i	$0.678280\pi$
$854$	−9.43483e6	−0.442680
$855$	0	0
$856$	−4.99824e6	−0.233149
$857$	−1.78039e7	−0.828063	−0.414032	−	0.910262i	$-0.635880\pi$
$857$	−1.78039e7	−0.828063	−0.414032	+	0.910262i	$0.635880\pi$
$858$	5.82826e6	0.270284
$859$	−302900.	−0.0140061	−0.00700304	−	0.999975i	$-0.502229\pi$
$859$	−302900.	−0.0140061	−0.00700304	+	0.999975i	$0.502229\pi$
$860$	0	0
$861$	−1.92432e7	−0.884647
$862$	−5.48650e6	−0.251494
$863$	4.37306e6	0.199875	0.0999374	−	0.994994i	$-0.468136\pi$
$863$	4.37306e6	0.199875	0.0999374	+	0.994994i	$0.468136\pi$
$864$	3.75581e6	0.171167
$865$	0	0
$866$	−1.21165e7	−0.549011
$867$	9.17988e6	0.414753
$868$	−2.68773e7	−1.21084
$869$	2.24861e7	1.01010
$870$	0	0
$871$	3.95959e6	0.176850
$872$	−1.25844e7	−0.560456
$873$	2.71366e6	0.120509
$874$	1.15104e6	0.0509697
$875$	0	0
$876$	−1.77726e7	−0.782509
$877$	−2.24189e7	−0.984272	−0.492136	−	0.870518i	$-0.663784\pi$
$877$	−2.24189e7	−0.984272	−0.492136	+	0.870518i	$0.663784\pi$
$878$	1.04209e7	0.456213
$879$	507654.	0.0221613
$880$	0	0
$881$	−1.40470e7	−0.609739	−0.304869	−	0.952394i	$-0.598613\pi$
$881$	−1.40470e7	−0.609739	−0.304869	+	0.952394i	$0.598613\pi$
$882$	99954.0	0.00432642
$883$	2.41189e7	1.04101	0.520505	−	0.853858i	$-0.325744\pi$
$883$	2.41189e7	1.04101	0.520505	+	0.853858i	$0.325744\pi$
$884$	−3.00029e7	−1.29132
$885$	0	0
$886$	−2.34497e6	−0.100358
$887$	−2.05454e7	−0.876809	−0.438404	−	0.898778i	$-0.644456\pi$
$887$	−2.05454e7	−0.876809	−0.438404	+	0.898778i	$0.644456\pi$
$888$	−153360.	−0.00652649
$889$	7.83710e6	0.332584
$890$	0	0
$891$	3.09679e6	0.130683
$892$	−5.90901e6	−0.248658
$893$	4.04782e7	1.69861
$894$	−2.48526e6	−0.103999
$895$	0	0
$896$	−2.45837e7	−1.02300
$897$	−1.62994e6	−0.0676378
$898$	8.28722e6	0.342940
$899$	1.23624e6	0.0510157
$900$	0	0
$901$	−3.36049e7	−1.37908
$902$	−1.52909e7	−0.625773
$903$	1.22554e7	0.500159
$904$	−1.84495e7	−0.750869
$905$	0	0
$906$	−6.52334e6	−0.264028
$907$	7.37425e6	0.297646	0.148823	−	0.988864i	$-0.452452\pi$
$907$	7.37425e6	0.297646	0.148823	+	0.988864i	$0.452452\pi$
$908$	2.57832e7	1.03782
$909$	6.38944e6	0.256480
$910$	0	0
$911$	−2.23806e7	−0.893462	−0.446731	−	0.894668i	$-0.647412\pi$
$911$	−2.23806e7	−0.893462	−0.446731	+	0.894668i	$0.647412\pi$
$912$	−1.28707e7	−0.512408
$913$	−3.49299e7	−1.38682
$914$	1.08268e7	0.428680
$915$	0	0
$916$	1.99164e6	0.0784282
$917$	−1.30427e7	−0.512204
$918$	2.27740e6	0.0891932
$919$	3.04744e7	1.19027	0.595136	−	0.803625i	$-0.297098\pi$
$919$	3.04744e7	1.19027	0.595136	+	0.803625i	$0.297098\pi$
$920$	0	0
$921$	1.58455e7	0.615541
$922$	1.07336e7	0.415832
$923$	−4.73944e7	−1.83114
$924$	−1.57006e7	−0.604974
$925$	0	0
$926$	1.06657e7	0.408753
$927$	6.64492e6	0.253975
$928$	875840.	0.0333853
$929$	1.87929e7	0.714421	0.357211	−	0.934024i	$-0.383728\pi$
$929$	1.87929e7	0.714421	0.357211	+	0.934024i	$0.383728\pi$
$930$	0	0
$931$	−1.34506e6	−0.0508590
$932$	2.80138e7	1.05641
$933$	−1.54250e7	−0.580124
$934$	−1.11366e7	−0.417721
$935$	0	0
$936$	−6.66792e6	−0.248772
$937$	9.77386e6	0.363678	0.181839	−	0.983328i	$-0.441795\pi$
$937$	9.77386e6	0.363678	0.181839	+	0.983328i	$0.441795\pi$
$938$	1.52381e6	0.0565488
$939$	1.28589e7	0.475926
$940$	0	0
$941$	1.24022e7	0.456590	0.228295	−	0.973592i	$-0.426685\pi$
$941$	1.24022e7	0.456590	0.228295	+	0.973592i	$0.426685\pi$
$942$	−4.42976e6	−0.162650
$943$	4.27627e6	0.156598
$944$	2.26976e7	0.828991
$945$	0	0
$946$	9.73830e6	0.353798
$947$	1.77052e7	0.641542	0.320771	−	0.947157i	$-0.396058\pi$
$947$	1.77052e7	0.641542	0.320771	+	0.947157i	$0.396058\pi$
$948$	−1.20053e7	−0.433862
$949$	4.83808e7	1.74384
$950$	0	0
$951$	6.69944e6	0.240208
$952$	−2.47421e7	−0.884797
$953$	−2.37213e6	−0.0846071	−0.0423036	−	0.999105i	$-0.513470\pi$
$953$	−2.37213e6	−0.0846071	−0.0423036	+	0.999105i	$0.513470\pi$
$954$	−3.48527e6	−0.123984
$955$	0	0
$956$	1.54146e7	0.545490
$957$	722160.	0.0254890
$958$	−3.14184e6	−0.110604
$959$	−4.05132e7	−1.42249
$960$	0	0
$961$	2.42528e7	0.847138
$962$	194824.	0.00678742
$963$	3.37381e6	0.117234
$964$	−1.52231e7	−0.527607
$965$	0	0
$966$	−627264.	−0.0216276
$967$	−567548.	−0.0195180	−0.00975902	−	0.999952i	$-0.503106\pi$
$967$	−567548.	−0.0195180	−0.00975902	+	0.999952i	$0.503106\pi$
$968$	−7.40796e6	−0.254103
$969$	−3.06464e7	−1.04851
$970$	0	0
$971$	−2.33139e7	−0.793536	−0.396768	−	0.917919i	$-0.629868\pi$
$971$	−2.33139e7	−0.793536	−0.396768	+	0.917919i	$0.629868\pi$
$972$	−1.65337e6	−0.0561313
$973$	−4.86024e6	−0.164579
$974$	1.54899e7	0.523182
$975$	0	0
$976$	−2.34441e7	−0.787788
$977$	−3.94860e6	−0.132345	−0.0661723	−	0.997808i	$-0.521079\pi$
$977$	−3.94860e6	−0.132345	−0.0661723	+	0.997808i	$0.521079\pi$
$978$	−3.06151e6	−0.102350
$979$	−4.24942e7	−1.41701
$980$	0	0
$981$	8.49447e6	0.281815
$982$	3.95430e6	0.130855
$983$	−5.01417e7	−1.65506	−0.827532	−	0.561418i	$-0.810256\pi$
$983$	−5.01417e7	−1.65506	−0.827532	+	0.561418i	$0.810256\pi$
$984$	1.74938e7	0.575967
$985$	0	0
$986$	531080.	0.0173967
$987$	−2.20588e7	−0.720757
$988$	4.18734e7	1.36473
$989$	−2.72342e6	−0.0885369
$990$	0	0
$991$	3.48675e6	0.112781	0.0563906	−	0.998409i	$-0.482041\pi$
$991$	3.48675e6	0.112781	0.0563906	+	0.998409i	$0.482041\pi$
$992$	3.74653e7	1.20879
$993$	−1.23217e6	−0.0396550
$994$	−1.82392e7	−0.585518
$995$	0	0
$996$	1.86490e7	0.595672
$997$	1.32373e7	0.421756	0.210878	−	0.977512i	$-0.432368\pi$
$997$	1.32373e7	0.421756	0.210878	+	0.977512i	$0.432368\pi$
$998$	−1.31199e7	−0.416969
$999$	103518.	0.00328172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.6.a.c.1.1		1
3.2	odd	2		225.6.a.c.1.1		1
5.2	odd	4		75.6.b.d.49.2		2
5.3	odd	4		75.6.b.d.49.1		2
5.4	even	2		15.6.a.a.1.1	✓	1
15.2	even	4		225.6.b.d.199.1		2
15.8	even	4		225.6.b.d.199.2		2
15.14	odd	2		45.6.a.c.1.1		1
20.19	odd	2		240.6.a.k.1.1		1
35.34	odd	2		735.6.a.a.1.1		1
40.19	odd	2		960.6.a.m.1.1		1
40.29	even	2		960.6.a.v.1.1		1
60.59	even	2		720.6.a.w.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.6.a.a.1.1	✓	1	5.4	even	2
45.6.a.c.1.1		1	15.14	odd	2
75.6.a.c.1.1		1	1.1	even	1	trivial
75.6.b.d.49.1		2	5.3	odd	4
75.6.b.d.49.2		2	5.2	odd	4
225.6.a.c.1.1		1	3.2	odd	2
225.6.b.d.199.1		2	15.2	even	4
225.6.b.d.199.2		2	15.8	even	4
240.6.a.k.1.1		1	20.19	odd	2
720.6.a.w.1.1		1	60.59	even	2
735.6.a.a.1.1		1	35.34	odd	2
960.6.a.m.1.1		1	40.19	odd	2
960.6.a.v.1.1		1	40.29	even	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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	75.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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