LMFDB - Embedded newform 1350.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(1,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1350.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.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:	$79.6525785077$
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	$\chi$	$=$	1350.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} -23.0000 q^{7} -8.00000 q^{8} +O(q^{10})$

$q-2.00000 q^{2} +4.00000 q^{4} -23.0000 q^{7} -8.00000 q^{8} -30.0000 q^{11} +34.0000 q^{13} +46.0000 q^{14} +16.0000 q^{16} -42.0000 q^{17} -139.000 q^{19} +60.0000 q^{22} +192.000 q^{23} -68.0000 q^{26} -92.0000 q^{28} -234.000 q^{29} -55.0000 q^{31} -32.0000 q^{32} +84.0000 q^{34} -191.000 q^{37} +278.000 q^{38} -138.000 q^{41} -53.0000 q^{43} -120.000 q^{44} -384.000 q^{46} +366.000 q^{47} +186.000 q^{49} +136.000 q^{52} -330.000 q^{53} +184.000 q^{56} +468.000 q^{58} +396.000 q^{59} +23.0000 q^{61} +110.000 q^{62} +64.0000 q^{64} -452.000 q^{67} -168.000 q^{68} -204.000 q^{71} +691.000 q^{73} +382.000 q^{74} -556.000 q^{76} +690.000 q^{77} -709.000 q^{79} +276.000 q^{82} +1098.00 q^{83} +106.000 q^{86} +240.000 q^{88} +816.000 q^{89} -782.000 q^{91} +768.000 q^{92} -732.000 q^{94} -905.000 q^{97} -372.000 q^{98} +O(q^{100})$

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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−23.0000	−1.24188	−0.620942	−	0.783857i	$-0.713250\pi$
$7$	−23.0000	−1.24188	−0.620942	+	0.783857i	$0.713250\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	−30.0000	−0.822304	−0.411152	−	0.911567i	$-0.634873\pi$
$11$	−30.0000	−0.822304	−0.411152	+	0.911567i	$0.634873\pi$
$12$	0	0
$13$	34.0000	0.725377	0.362689	−	0.931910i	$-0.381859\pi$
$13$	34.0000	0.725377	0.362689	+	0.931910i	$0.381859\pi$
$14$	46.0000	0.878144
$15$	0	0
$16$	16.0000	0.250000
$17$	−42.0000	−0.599206	−0.299603	−	0.954064i	$-0.596854\pi$
$17$	−42.0000	−0.599206	−0.299603	+	0.954064i	$0.596854\pi$
$18$	0	0
$19$	−139.000	−1.67836	−0.839179	−	0.543856i	$-0.816964\pi$
$19$	−139.000	−1.67836	−0.839179	+	0.543856i	$0.816964\pi$
$20$	0	0
$21$	0	0
$22$	60.0000	0.581456
$23$	192.000	1.74064	0.870321	−	0.492485i	$-0.163911\pi$
$23$	192.000	1.74064	0.870321	+	0.492485i	$0.163911\pi$
$24$	0	0
$25$	0	0
$26$	−68.0000	−0.512919
$27$	0	0
$28$	−92.0000	−0.620942
$29$	−234.000	−1.49837	−0.749185	−	0.662361i	$-0.769554\pi$
$29$	−234.000	−1.49837	−0.749185	+	0.662361i	$0.769554\pi$
$30$	0	0
$31$	−55.0000	−0.318655	−0.159327	−	0.987226i	$-0.550933\pi$
$31$	−55.0000	−0.318655	−0.159327	+	0.987226i	$0.550933\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	84.0000	0.423702
$35$	0	0
$36$	0	0
$37$	−191.000	−0.848654	−0.424327	−	0.905509i	$-0.639489\pi$
$37$	−191.000	−0.848654	−0.424327	+	0.905509i	$0.639489\pi$
$38$	278.000	1.18678
$39$	0	0
$40$	0	0
$41$	−138.000	−0.525658	−0.262829	−	0.964842i	$-0.584656\pi$
$41$	−138.000	−0.525658	−0.262829	+	0.964842i	$0.584656\pi$
$42$	0	0
$43$	−53.0000	−0.187963	−0.0939817	−	0.995574i	$-0.529960\pi$
$43$	−53.0000	−0.187963	−0.0939817	+	0.995574i	$0.529960\pi$
$44$	−120.000	−0.411152
$45$	0	0
$46$	−384.000	−1.23082
$47$	366.000	1.13588	0.567942	−	0.823068i	$-0.307740\pi$
$47$	366.000	1.13588	0.567942	+	0.823068i	$0.307740\pi$
$48$	0	0
$49$	186.000	0.542274
$50$	0	0
$51$	0	0
$52$	136.000	0.362689
$53$	−330.000	−0.855264	−0.427632	−	0.903953i	$-0.640652\pi$
$53$	−330.000	−0.855264	−0.427632	+	0.903953i	$0.640652\pi$
$54$	0	0
$55$	0	0
$56$	184.000	0.439072
$57$	0	0
$58$	468.000	1.05951
$59$	396.000	0.873810	0.436905	−	0.899508i	$-0.356075\pi$
$59$	396.000	0.873810	0.436905	+	0.899508i	$0.356075\pi$
$60$	0	0
$61$	23.0000	0.0482762	0.0241381	−	0.999709i	$-0.492316\pi$
$61$	23.0000	0.0482762	0.0241381	+	0.999709i	$0.492316\pi$
$62$	110.000	0.225323
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−452.000	−0.824188	−0.412094	−	0.911141i	$-0.635202\pi$
$67$	−452.000	−0.824188	−0.412094	+	0.911141i	$0.635202\pi$
$68$	−168.000	−0.299603
$69$	0	0
$70$	0	0
$71$	−204.000	−0.340991	−0.170495	−	0.985358i	$-0.554537\pi$
$71$	−204.000	−0.340991	−0.170495	+	0.985358i	$0.554537\pi$
$72$	0	0
$73$	691.000	1.10788	0.553941	−	0.832556i	$-0.313123\pi$
$73$	691.000	1.10788	0.553941	+	0.832556i	$0.313123\pi$
$74$	382.000	0.600089
$75$	0	0
$76$	−556.000	−0.839179
$77$	690.000	1.02121
$78$	0	0
$79$	−709.000	−1.00973	−0.504865	−	0.863198i	$-0.668458\pi$
$79$	−709.000	−1.00973	−0.504865	+	0.863198i	$0.668458\pi$
$80$	0	0
$81$	0	0
$82$	276.000	0.371696
$83$	1098.00	1.45206	0.726031	−	0.687662i	$-0.241363\pi$
$83$	1098.00	1.45206	0.726031	+	0.687662i	$0.241363\pi$
$84$	0	0
$85$	0	0
$86$	106.000	0.132910
$87$	0	0
$88$	240.000	0.290728
$89$	816.000	0.971863	0.485932	−	0.873997i	$-0.338480\pi$
$89$	816.000	0.971863	0.485932	+	0.873997i	$0.338480\pi$
$90$	0	0
$91$	−782.000	−0.900834
$92$	768.000	0.870321
$93$	0	0
$94$	−732.000	−0.803192
$95$	0	0
$96$	0	0
$97$	−905.000	−0.947308	−0.473654	−	0.880711i	$-0.657065\pi$
$97$	−905.000	−0.947308	−0.473654	+	0.880711i	$0.657065\pi$
$98$	−372.000	−0.383446
$99$	0	0
$100$	0	0
$101$	1278.00	1.25907	0.629533	−	0.776973i	$-0.283246\pi$
$101$	1278.00	1.25907	0.629533	+	0.776973i	$0.283246\pi$
$102$	0	0
$103$	−605.000	−0.578761	−0.289381	−	0.957214i	$-0.593449\pi$
$103$	−605.000	−0.578761	−0.289381	+	0.957214i	$0.593449\pi$
$104$	−272.000	−0.256460
$105$	0	0
$106$	660.000	0.604763
$107$	1488.00	1.34440	0.672198	−	0.740371i	$-0.265350\pi$
$107$	1488.00	1.34440	0.672198	+	0.740371i	$0.265350\pi$
$108$	0	0
$109$	593.000	0.521093	0.260546	−	0.965461i	$-0.416097\pi$
$109$	593.000	0.521093	0.260546	+	0.965461i	$0.416097\pi$
$110$	0	0
$111$	0	0
$112$	−368.000	−0.310471
$113$	324.000	0.269729	0.134864	−	0.990864i	$-0.456940\pi$
$113$	324.000	0.269729	0.134864	+	0.990864i	$0.456940\pi$
$114$	0	0
$115$	0	0
$116$	−936.000	−0.749185
$117$	0	0
$118$	−792.000	−0.617877
$119$	966.000	0.744143
$120$	0	0
$121$	−431.000	−0.323817
$122$	−46.0000	−0.0341364
$123$	0	0
$124$	−220.000	−0.159327
$125$	0	0
$126$	0	0
$127$	−1928.00	−1.34711	−0.673553	−	0.739139i	$-0.735232\pi$
$127$	−1928.00	−1.34711	−0.673553	+	0.739139i	$0.735232\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−2742.00	−1.82878	−0.914388	−	0.404839i	$-0.867328\pi$
$131$	−2742.00	−1.82878	−0.914388	+	0.404839i	$0.867328\pi$
$132$	0	0
$133$	3197.00	2.08432
$134$	904.000	0.582789
$135$	0	0
$136$	336.000	0.211851
$137$	1326.00	0.826918	0.413459	−	0.910523i	$-0.364320\pi$
$137$	1326.00	0.826918	0.413459	+	0.910523i	$0.364320\pi$
$138$	0	0
$139$	893.000	0.544916	0.272458	−	0.962168i	$-0.412163\pi$
$139$	893.000	0.544916	0.272458	+	0.962168i	$0.412163\pi$
$140$	0	0
$141$	0	0
$142$	408.000	0.241117
$143$	−1020.00	−0.596480
$144$	0	0
$145$	0	0
$146$	−1382.00	−0.783391
$147$	0	0
$148$	−764.000	−0.424327
$149$	2502.00	1.37565	0.687825	−	0.725877i	$-0.258566\pi$
$149$	2502.00	1.37565	0.687825	+	0.725877i	$0.258566\pi$
$150$	0	0
$151$	−2767.00	−1.49123	−0.745613	−	0.666379i	$-0.767843\pi$
$151$	−2767.00	−1.49123	−0.745613	+	0.666379i	$0.767843\pi$
$152$	1112.00	0.593389
$153$	0	0
$154$	−1380.00	−0.722101
$155$	0	0
$156$	0	0
$157$	2701.00	1.37301	0.686507	−	0.727123i	$-0.259143\pi$
$157$	2701.00	1.37301	0.686507	+	0.727123i	$0.259143\pi$
$158$	1418.00	0.713987
$159$	0	0
$160$	0	0
$161$	−4416.00	−2.16167
$162$	0	0
$163$	−1748.00	−0.839963	−0.419981	−	0.907533i	$-0.637963\pi$
$163$	−1748.00	−0.839963	−0.419981	+	0.907533i	$0.637963\pi$
$164$	−552.000	−0.262829
$165$	0	0
$166$	−2196.00	−1.02676
$167$	−534.000	−0.247438	−0.123719	−	0.992317i	$-0.539482\pi$
$167$	−534.000	−0.247438	−0.123719	+	0.992317i	$0.539482\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	0	0
$172$	−212.000	−0.0939817
$173$	−192.000	−0.0843786	−0.0421893	−	0.999110i	$-0.513433\pi$
$173$	−192.000	−0.0843786	−0.0421893	+	0.999110i	$0.513433\pi$
$174$	0	0
$175$	0	0
$176$	−480.000	−0.205576
$177$	0	0
$178$	−1632.00	−0.687211
$179$	−1140.00	−0.476020	−0.238010	−	0.971263i	$-0.576495\pi$
$179$	−1140.00	−0.476020	−0.238010	+	0.971263i	$0.576495\pi$
$180$	0	0
$181$	398.000	0.163443	0.0817213	−	0.996655i	$-0.473958\pi$
$181$	398.000	0.163443	0.0817213	+	0.996655i	$0.473958\pi$
$182$	1564.00	0.636986
$183$	0	0
$184$	−1536.00	−0.615410
$185$	0	0
$186$	0	0
$187$	1260.00	0.492729
$188$	1464.00	0.567942
$189$	0	0
$190$	0	0
$191$	−3474.00	−1.31607	−0.658036	−	0.752986i	$-0.728613\pi$
$191$	−3474.00	−1.31607	−0.658036	+	0.752986i	$0.728613\pi$
$192$	0	0
$193$	2713.00	1.01184	0.505922	−	0.862579i	$-0.331152\pi$
$193$	2713.00	1.01184	0.505922	+	0.862579i	$0.331152\pi$
$194$	1810.00	0.669848
$195$	0	0
$196$	744.000	0.271137
$197$	4734.00	1.71210	0.856050	−	0.516894i	$-0.172912\pi$
$197$	4734.00	1.71210	0.856050	+	0.516894i	$0.172912\pi$
$198$	0	0
$199$	5132.00	1.82813	0.914065	−	0.405568i	$-0.132926\pi$
$199$	5132.00	1.82813	0.914065	+	0.405568i	$0.132926\pi$
$200$	0	0
$201$	0	0
$202$	−2556.00	−0.890295
$203$	5382.00	1.86080
$204$	0	0
$205$	0	0
$206$	1210.00	0.409246
$207$	0	0
$208$	544.000	0.181344
$209$	4170.00	1.38012
$210$	0	0
$211$	5240.00	1.70965	0.854826	−	0.518915i	$-0.173664\pi$
$211$	5240.00	1.70965	0.854826	+	0.518915i	$0.173664\pi$
$212$	−1320.00	−0.427632
$213$	0	0
$214$	−2976.00	−0.950632
$215$	0	0
$216$	0	0
$217$	1265.00	0.395732
$218$	−1186.00	−0.368468
$219$	0	0
$220$	0	0
$221$	−1428.00	−0.434650
$222$	0	0
$223$	4519.00	1.35702	0.678508	−	0.734593i	$-0.262627\pi$
$223$	4519.00	1.35702	0.678508	+	0.734593i	$0.262627\pi$
$224$	736.000	0.219536
$225$	0	0
$226$	−648.000	−0.190727
$227$	5064.00	1.48066	0.740329	−	0.672244i	$-0.234670\pi$
$227$	5064.00	1.48066	0.740329	+	0.672244i	$0.234670\pi$
$228$	0	0
$229$	2573.00	0.742483	0.371242	−	0.928536i	$-0.378932\pi$
$229$	2573.00	0.742483	0.371242	+	0.928536i	$0.378932\pi$
$230$	0	0
$231$	0	0
$232$	1872.00	0.529754
$233$	4098.00	1.15223	0.576114	−	0.817370i	$-0.304569\pi$
$233$	4098.00	1.15223	0.576114	+	0.817370i	$0.304569\pi$
$234$	0	0
$235$	0	0
$236$	1584.00	0.436905
$237$	0	0
$238$	−1932.00	−0.526189
$239$	306.000	0.0828180	0.0414090	−	0.999142i	$-0.486815\pi$
$239$	306.000	0.0828180	0.0414090	+	0.999142i	$0.486815\pi$
$240$	0	0
$241$	6482.00	1.73254	0.866270	−	0.499575i	$-0.166511\pi$
$241$	6482.00	1.73254	0.866270	+	0.499575i	$0.166511\pi$
$242$	862.000	0.228973
$243$	0	0
$244$	92.0000	0.0241381
$245$	0	0
$246$	0	0
$247$	−4726.00	−1.21744
$248$	440.000	0.112661
$249$	0	0
$250$	0	0
$251$	5850.00	1.47111	0.735555	−	0.677465i	$-0.236921\pi$
$251$	5850.00	1.47111	0.735555	+	0.677465i	$0.236921\pi$
$252$	0	0
$253$	−5760.00	−1.43134
$254$	3856.00	0.952547
$255$	0	0
$256$	256.000	0.0625000
$257$	5598.00	1.35873	0.679365	−	0.733800i	$-0.262255\pi$
$257$	5598.00	1.35873	0.679365	+	0.733800i	$0.262255\pi$
$258$	0	0
$259$	4393.00	1.05393
$260$	0	0
$261$	0	0
$262$	5484.00	1.29314
$263$	−8286.00	−1.94272	−0.971362	−	0.237603i	$-0.923638\pi$
$263$	−8286.00	−1.94272	−0.971362	+	0.237603i	$0.923638\pi$
$264$	0	0
$265$	0	0
$266$	−6394.00	−1.47384
$267$	0	0
$268$	−1808.00	−0.412094
$269$	504.000	0.114236	0.0571179	−	0.998367i	$-0.481809\pi$
$269$	504.000	0.114236	0.0571179	+	0.998367i	$0.481809\pi$
$270$	0	0
$271$	−4489.00	−1.00623	−0.503113	−	0.864221i	$-0.667812\pi$
$271$	−4489.00	−1.00623	−0.503113	+	0.864221i	$0.667812\pi$
$272$	−672.000	−0.149801
$273$	0	0
$274$	−2652.00	−0.584720
$275$	0	0
$276$	0	0
$277$	−2213.00	−0.480023	−0.240011	−	0.970770i	$-0.577151\pi$
$277$	−2213.00	−0.480023	−0.240011	+	0.970770i	$0.577151\pi$
$278$	−1786.00	−0.385314
$279$	0	0
$280$	0	0
$281$	2718.00	0.577019	0.288509	−	0.957477i	$-0.406840\pi$
$281$	2718.00	0.577019	0.288509	+	0.957477i	$0.406840\pi$
$282$	0	0
$283$	−5615.00	−1.17942	−0.589712	−	0.807613i	$-0.700759\pi$
$283$	−5615.00	−1.17942	−0.589712	+	0.807613i	$0.700759\pi$
$284$	−816.000	−0.170495
$285$	0	0
$286$	2040.00	0.421775
$287$	3174.00	0.652806
$288$	0	0
$289$	−3149.00	−0.640953
$290$	0	0
$291$	0	0
$292$	2764.00	0.553941
$293$	1488.00	0.296689	0.148345	−	0.988936i	$-0.452606\pi$
$293$	1488.00	0.296689	0.148345	+	0.988936i	$0.452606\pi$
$294$	0	0
$295$	0	0
$296$	1528.00	0.300045
$297$	0	0
$298$	−5004.00	−0.972731
$299$	6528.00	1.26262
$300$	0	0
$301$	1219.00	0.233429
$302$	5534.00	1.05446
$303$	0	0
$304$	−2224.00	−0.419589
$305$	0	0
$306$	0	0
$307$	−7487.00	−1.39188	−0.695938	−	0.718102i	$-0.745011\pi$
$307$	−7487.00	−1.39188	−0.695938	+	0.718102i	$0.745011\pi$
$308$	2760.00	0.510603
$309$	0	0
$310$	0	0
$311$	−8118.00	−1.48016	−0.740080	−	0.672519i	$-0.765212\pi$
$311$	−8118.00	−1.48016	−0.740080	+	0.672519i	$0.765212\pi$
$312$	0	0
$313$	5002.00	0.903290	0.451645	−	0.892198i	$-0.350837\pi$
$313$	5002.00	0.903290	0.451645	+	0.892198i	$0.350837\pi$
$314$	−5402.00	−0.970868
$315$	0	0
$316$	−2836.00	−0.504865
$317$	5082.00	0.900421	0.450211	−	0.892922i	$-0.351349\pi$
$317$	5082.00	0.900421	0.450211	+	0.892922i	$0.351349\pi$
$318$	0	0
$319$	7020.00	1.23211
$320$	0	0
$321$	0	0
$322$	8832.00	1.52853
$323$	5838.00	1.00568
$324$	0	0
$325$	0	0
$326$	3496.00	0.593943
$327$	0	0
$328$	1104.00	0.185848
$329$	−8418.00	−1.41064
$330$	0	0
$331$	7625.00	1.26619	0.633094	−	0.774075i	$-0.281785\pi$
$331$	7625.00	1.26619	0.633094	+	0.774075i	$0.281785\pi$
$332$	4392.00	0.726031
$333$	0	0
$334$	1068.00	0.174965
$335$	0	0
$336$	0	0
$337$	3778.00	0.610685	0.305342	−	0.952243i	$-0.401229\pi$
$337$	3778.00	0.610685	0.305342	+	0.952243i	$0.401229\pi$
$338$	2082.00	0.335047
$339$	0	0
$340$	0	0
$341$	1650.00	0.262031
$342$	0	0
$343$	3611.00	0.568442
$344$	424.000	0.0664551
$345$	0	0
$346$	384.000	0.0596646
$347$	−8268.00	−1.27911	−0.639553	−	0.768747i	$-0.720880\pi$
$347$	−8268.00	−1.27911	−0.639553	+	0.768747i	$0.720880\pi$
$348$	0	0
$349$	1379.00	0.211508	0.105754	−	0.994392i	$-0.466274\pi$
$349$	1379.00	0.211508	0.105754	+	0.994392i	$0.466274\pi$
$350$	0	0
$351$	0	0
$352$	960.000	0.145364
$353$	3072.00	0.463190	0.231595	−	0.972812i	$-0.425606\pi$
$353$	3072.00	0.463190	0.231595	+	0.972812i	$0.425606\pi$
$354$	0	0
$355$	0	0
$356$	3264.00	0.485932
$357$	0	0
$358$	2280.00	0.336597
$359$	7446.00	1.09467	0.547333	−	0.836915i	$-0.315643\pi$
$359$	7446.00	1.09467	0.547333	+	0.836915i	$0.315643\pi$
$360$	0	0
$361$	12462.0	1.81688
$362$	−796.000	−0.115571
$363$	0	0
$364$	−3128.00	−0.450417
$365$	0	0
$366$	0	0
$367$	6496.00	0.923947	0.461973	−	0.886894i	$-0.347142\pi$
$367$	6496.00	0.923947	0.461973	+	0.886894i	$0.347142\pi$
$368$	3072.00	0.435161
$369$	0	0
$370$	0	0
$371$	7590.00	1.06214
$372$	0	0
$373$	1633.00	0.226685	0.113343	−	0.993556i	$-0.463844\pi$
$373$	1633.00	0.226685	0.113343	+	0.993556i	$0.463844\pi$
$374$	−2520.00	−0.348412
$375$	0	0
$376$	−2928.00	−0.401596
$377$	−7956.00	−1.08688
$378$	0	0
$379$	6788.00	0.919990	0.459995	−	0.887922i	$-0.347851\pi$
$379$	6788.00	0.919990	0.459995	+	0.887922i	$0.347851\pi$
$380$	0	0
$381$	0	0
$382$	6948.00	0.930604
$383$	6582.00	0.878132	0.439066	−	0.898455i	$-0.355309\pi$
$383$	6582.00	0.878132	0.439066	+	0.898455i	$0.355309\pi$
$384$	0	0
$385$	0	0
$386$	−5426.00	−0.715482
$387$	0	0
$388$	−3620.00	−0.473654
$389$	−2850.00	−0.371467	−0.185734	−	0.982600i	$-0.559466\pi$
$389$	−2850.00	−0.371467	−0.185734	+	0.982600i	$0.559466\pi$
$390$	0	0
$391$	−8064.00	−1.04300
$392$	−1488.00	−0.191723
$393$	0	0
$394$	−9468.00	−1.21064
$395$	0	0
$396$	0	0
$397$	−7451.00	−0.941952	−0.470976	−	0.882146i	$-0.656098\pi$
$397$	−7451.00	−0.941952	−0.470976	+	0.882146i	$0.656098\pi$
$398$	−10264.0	−1.29268
$399$	0	0
$400$	0	0
$401$	−14124.0	−1.75890	−0.879450	−	0.475991i	$-0.842089\pi$
$401$	−14124.0	−1.75890	−0.879450	+	0.475991i	$0.842089\pi$
$402$	0	0
$403$	−1870.00	−0.231145
$404$	5112.00	0.629533
$405$	0	0
$406$	−10764.0	−1.31578
$407$	5730.00	0.697851
$408$	0	0
$409$	6374.00	0.770597	0.385298	−	0.922792i	$-0.374099\pi$
$409$	6374.00	0.770597	0.385298	+	0.922792i	$0.374099\pi$
$410$	0	0
$411$	0	0
$412$	−2420.00	−0.289381
$413$	−9108.00	−1.08517
$414$	0	0
$415$	0	0
$416$	−1088.00	−0.128230
$417$	0	0
$418$	−8340.00	−0.975892
$419$	−3948.00	−0.460316	−0.230158	−	0.973153i	$-0.573924\pi$
$419$	−3948.00	−0.460316	−0.230158	+	0.973153i	$0.573924\pi$
$420$	0	0
$421$	12629.0	1.46199	0.730997	−	0.682380i	$-0.239055\pi$
$421$	12629.0	1.46199	0.730997	+	0.682380i	$0.239055\pi$
$422$	−10480.0	−1.20891
$423$	0	0
$424$	2640.00	0.302381
$425$	0	0
$426$	0	0
$427$	−529.000	−0.0599534
$428$	5952.00	0.672198
$429$	0	0
$430$	0	0
$431$	−4842.00	−0.541139	−0.270570	−	0.962700i	$-0.587212\pi$
$431$	−4842.00	−0.541139	−0.270570	+	0.962700i	$0.587212\pi$
$432$	0	0
$433$	−3851.00	−0.427407	−0.213704	−	0.976899i	$-0.568553\pi$
$433$	−3851.00	−0.427407	−0.213704	+	0.976899i	$0.568553\pi$
$434$	−2530.00	−0.279825
$435$	0	0
$436$	2372.00	0.260546
$437$	−26688.0	−2.92142
$438$	0	0
$439$	−7435.00	−0.808322	−0.404161	−	0.914688i	$-0.632436\pi$
$439$	−7435.00	−0.808322	−0.404161	+	0.914688i	$0.632436\pi$
$440$	0	0
$441$	0	0
$442$	2856.00	0.307344
$443$	−5760.00	−0.617756	−0.308878	−	0.951102i	$-0.599953\pi$
$443$	−5760.00	−0.617756	−0.308878	+	0.951102i	$0.599953\pi$
$444$	0	0
$445$	0	0
$446$	−9038.00	−0.959555
$447$	0	0
$448$	−1472.00	−0.155235
$449$	2190.00	0.230184	0.115092	−	0.993355i	$-0.463284\pi$
$449$	2190.00	0.230184	0.115092	+	0.993355i	$0.463284\pi$
$450$	0	0
$451$	4140.00	0.432251
$452$	1296.00	0.134864
$453$	0	0
$454$	−10128.0	−1.04698
$455$	0	0
$456$	0	0
$457$	−7202.00	−0.737189	−0.368594	−	0.929590i	$-0.620161\pi$
$457$	−7202.00	−0.737189	−0.368594	+	0.929590i	$0.620161\pi$
$458$	−5146.00	−0.525015
$459$	0	0
$460$	0	0
$461$	13476.0	1.36147	0.680737	−	0.732528i	$-0.261659\pi$
$461$	13476.0	1.36147	0.680737	+	0.732528i	$0.261659\pi$
$462$	0	0
$463$	10843.0	1.08837	0.544187	−	0.838964i	$-0.316838\pi$
$463$	10843.0	1.08837	0.544187	+	0.838964i	$0.316838\pi$
$464$	−3744.00	−0.374592
$465$	0	0
$466$	−8196.00	−0.814748
$467$	6108.00	0.605235	0.302617	−	0.953112i	$-0.402140\pi$
$467$	6108.00	0.605235	0.302617	+	0.953112i	$0.402140\pi$
$468$	0	0
$469$	10396.0	1.02355
$470$	0	0
$471$	0	0
$472$	−3168.00	−0.308939
$473$	1590.00	0.154563
$474$	0	0
$475$	0	0
$476$	3864.00	0.372072
$477$	0	0
$478$	−612.000	−0.0585611
$479$	−9852.00	−0.939769	−0.469885	−	0.882728i	$-0.655704\pi$
$479$	−9852.00	−0.939769	−0.469885	+	0.882728i	$0.655704\pi$
$480$	0	0
$481$	−6494.00	−0.615594
$482$	−12964.0	−1.22509
$483$	0	0
$484$	−1724.00	−0.161908
$485$	0	0
$486$	0	0
$487$	−7796.00	−0.725401	−0.362701	−	0.931906i	$-0.618145\pi$
$487$	−7796.00	−0.725401	−0.362701	+	0.931906i	$0.618145\pi$
$488$	−184.000	−0.0170682
$489$	0	0
$490$	0	0
$491$	−2454.00	−0.225555	−0.112777	−	0.993620i	$-0.535975\pi$
$491$	−2454.00	−0.225555	−0.112777	+	0.993620i	$0.535975\pi$
$492$	0	0
$493$	9828.00	0.897831
$494$	9452.00	0.860862
$495$	0	0
$496$	−880.000	−0.0796636
$497$	4692.00	0.423471
$498$	0	0
$499$	−2953.00	−0.264919	−0.132459	−	0.991188i	$-0.542287\pi$
$499$	−2953.00	−0.264919	−0.132459	+	0.991188i	$0.542287\pi$
$500$	0	0
$501$	0	0
$502$	−11700.0	−1.04023
$503$	−11322.0	−1.00362	−0.501812	−	0.864977i	$-0.667333\pi$
$503$	−11322.0	−1.00362	−0.501812	+	0.864977i	$0.667333\pi$
$504$	0	0
$505$	0	0
$506$	11520.0	1.01211
$507$	0	0
$508$	−7712.00	−0.673553
$509$	−9696.00	−0.844337	−0.422169	−	0.906517i	$-0.638731\pi$
$509$	−9696.00	−0.844337	−0.422169	+	0.906517i	$0.638731\pi$
$510$	0	0
$511$	−15893.0	−1.37586
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−11196.0	−0.960767
$515$	0	0
$516$	0	0
$517$	−10980.0	−0.934042
$518$	−8786.00	−0.745241
$519$	0	0
$520$	0	0
$521$	12192.0	1.02522	0.512612	−	0.858621i	$-0.328678\pi$
$521$	12192.0	1.02522	0.512612	+	0.858621i	$0.328678\pi$
$522$	0	0
$523$	8491.00	0.709915	0.354957	−	0.934882i	$-0.384495\pi$
$523$	8491.00	0.709915	0.354957	+	0.934882i	$0.384495\pi$
$524$	−10968.0	−0.914388
$525$	0	0
$526$	16572.0	1.37371
$527$	2310.00	0.190940
$528$	0	0
$529$	24697.0	2.02983
$530$	0	0
$531$	0	0
$532$	12788.0	1.04216
$533$	−4692.00	−0.381300
$534$	0	0
$535$	0	0
$536$	3616.00	0.291394
$537$	0	0
$538$	−1008.00	−0.0807769
$539$	−5580.00	−0.445914
$540$	0	0
$541$	−9355.00	−0.743443	−0.371722	−	0.928344i	$-0.621232\pi$
$541$	−9355.00	−0.743443	−0.371722	+	0.928344i	$0.621232\pi$
$542$	8978.00	0.711509
$543$	0	0
$544$	1344.00	0.105926
$545$	0	0
$546$	0	0
$547$	295.000	0.0230590	0.0115295	−	0.999934i	$-0.496330\pi$
$547$	295.000	0.0230590	0.0115295	+	0.999934i	$0.496330\pi$
$548$	5304.00	0.413459
$549$	0	0
$550$	0	0
$551$	32526.0	2.51480
$552$	0	0
$553$	16307.0	1.25397
$554$	4426.00	0.339427
$555$	0	0
$556$	3572.00	0.272458
$557$	−16914.0	−1.28666	−0.643330	−	0.765589i	$-0.722448\pi$
$557$	−16914.0	−1.28666	−0.643330	+	0.765589i	$0.722448\pi$
$558$	0	0
$559$	−1802.00	−0.136344
$560$	0	0
$561$	0	0
$562$	−5436.00	−0.408014
$563$	−12108.0	−0.906379	−0.453189	−	0.891414i	$-0.649714\pi$
$563$	−12108.0	−0.906379	−0.453189	+	0.891414i	$0.649714\pi$
$564$	0	0
$565$	0	0
$566$	11230.0	0.833979
$567$	0	0
$568$	1632.00	0.120558
$569$	6960.00	0.512792	0.256396	−	0.966572i	$-0.417465\pi$
$569$	6960.00	0.512792	0.256396	+	0.966572i	$0.417465\pi$
$570$	0	0
$571$	−10687.0	−0.783252	−0.391626	−	0.920124i	$-0.628087\pi$
$571$	−10687.0	−0.783252	−0.391626	+	0.920124i	$0.628087\pi$
$572$	−4080.00	−0.298240
$573$	0	0
$574$	−6348.00	−0.461603
$575$	0	0
$576$	0	0
$577$	8329.00	0.600937	0.300469	−	0.953792i	$-0.402857\pi$
$577$	8329.00	0.600937	0.300469	+	0.953792i	$0.402857\pi$
$578$	6298.00	0.453222
$579$	0	0
$580$	0	0
$581$	−25254.0	−1.80329
$582$	0	0
$583$	9900.00	0.703287
$584$	−5528.00	−0.391696
$585$	0	0
$586$	−2976.00	−0.209791
$587$	−25302.0	−1.77909	−0.889545	−	0.456848i	$-0.848978\pi$
$587$	−25302.0	−1.77909	−0.889545	+	0.456848i	$0.848978\pi$
$588$	0	0
$589$	7645.00	0.534816
$590$	0	0
$591$	0	0
$592$	−3056.00	−0.212164
$593$	−22248.0	−1.54067	−0.770334	−	0.637641i	$-0.779910\pi$
$593$	−22248.0	−1.54067	−0.770334	+	0.637641i	$0.779910\pi$
$594$	0	0
$595$	0	0
$596$	10008.0	0.687825
$597$	0	0
$598$	−13056.0	−0.892809
$599$	−24252.0	−1.65427	−0.827137	−	0.562001i	$-0.810032\pi$
$599$	−24252.0	−1.65427	−0.827137	+	0.562001i	$0.810032\pi$
$600$	0	0
$601$	−9829.00	−0.667110	−0.333555	−	0.942731i	$-0.608248\pi$
$601$	−9829.00	−0.667110	−0.333555	+	0.942731i	$0.608248\pi$
$602$	−2438.00	−0.165059
$603$	0	0
$604$	−11068.0	−0.745613
$605$	0	0
$606$	0	0
$607$	14155.0	0.946514	0.473257	−	0.880925i	$-0.343078\pi$
$607$	14155.0	0.946514	0.473257	+	0.880925i	$0.343078\pi$
$608$	4448.00	0.296694
$609$	0	0
$610$	0	0
$611$	12444.0	0.823945
$612$	0	0
$613$	−23051.0	−1.51879	−0.759397	−	0.650627i	$-0.774506\pi$
$613$	−23051.0	−1.51879	−0.759397	+	0.650627i	$0.774506\pi$
$614$	14974.0	0.984204
$615$	0	0
$616$	−5520.00	−0.361051
$617$	−8352.00	−0.544958	−0.272479	−	0.962162i	$-0.587843\pi$
$617$	−8352.00	−0.544958	−0.272479	+	0.962162i	$0.587843\pi$
$618$	0	0
$619$	−24331.0	−1.57988	−0.789940	−	0.613184i	$-0.789888\pi$
$619$	−24331.0	−1.57988	−0.789940	+	0.613184i	$0.789888\pi$
$620$	0	0
$621$	0	0
$622$	16236.0	1.04663
$623$	−18768.0	−1.20694
$624$	0	0
$625$	0	0
$626$	−10004.0	−0.638722
$627$	0	0
$628$	10804.0	0.686507
$629$	8022.00	0.508518
$630$	0	0
$631$	2216.00	0.139806	0.0699030	−	0.997554i	$-0.477731\pi$
$631$	2216.00	0.139806	0.0699030	+	0.997554i	$0.477731\pi$
$632$	5672.00	0.356994
$633$	0	0
$634$	−10164.0	−0.636694
$635$	0	0
$636$	0	0
$637$	6324.00	0.393353
$638$	−14040.0	−0.871237
$639$	0	0
$640$	0	0
$641$	28500.0	1.75613	0.878067	−	0.478537i	$-0.158833\pi$
$641$	28500.0	1.75613	0.878067	+	0.478537i	$0.158833\pi$
$642$	0	0
$643$	25252.0	1.54874	0.774371	−	0.632731i	$-0.218066\pi$
$643$	25252.0	1.54874	0.774371	+	0.632731i	$0.218066\pi$
$644$	−17664.0	−1.08084
$645$	0	0
$646$	−11676.0	−0.711124
$647$	−28920.0	−1.75728	−0.878642	−	0.477481i	$-0.841550\pi$
$647$	−28920.0	−1.75728	−0.878642	+	0.477481i	$0.841550\pi$
$648$	0	0
$649$	−11880.0	−0.718537
$650$	0	0
$651$	0	0
$652$	−6992.00	−0.419981
$653$	32268.0	1.93376	0.966879	−	0.255234i	$-0.0821525\pi$
$653$	32268.0	1.93376	0.966879	+	0.255234i	$0.0821525\pi$
$654$	0	0
$655$	0	0
$656$	−2208.00	−0.131415
$657$	0	0
$658$	16836.0	0.997471
$659$	10050.0	0.594070	0.297035	−	0.954867i	$-0.404002\pi$
$659$	10050.0	0.594070	0.297035	+	0.954867i	$0.404002\pi$
$660$	0	0
$661$	−4561.00	−0.268385	−0.134192	−	0.990955i	$-0.542844\pi$
$661$	−4561.00	−0.268385	−0.134192	+	0.990955i	$0.542844\pi$
$662$	−15250.0	−0.895329
$663$	0	0
$664$	−8784.00	−0.513381
$665$	0	0
$666$	0	0
$667$	−44928.0	−2.60812
$668$	−2136.00	−0.123719
$669$	0	0
$670$	0	0
$671$	−690.000	−0.0396977
$672$	0	0
$673$	−21359.0	−1.22337	−0.611686	−	0.791101i	$-0.709508\pi$
$673$	−21359.0	−1.22337	−0.611686	+	0.791101i	$0.709508\pi$
$674$	−7556.00	−0.431819
$675$	0	0
$676$	−4164.00	−0.236914
$677$	15042.0	0.853931	0.426965	−	0.904268i	$-0.359583\pi$
$677$	15042.0	0.853931	0.426965	+	0.904268i	$0.359583\pi$
$678$	0	0
$679$	20815.0	1.17645
$680$	0	0
$681$	0	0
$682$	−3300.00	−0.185284
$683$	27462.0	1.53851	0.769256	−	0.638940i	$-0.220627\pi$
$683$	27462.0	1.53851	0.769256	+	0.638940i	$0.220627\pi$
$684$	0	0
$685$	0	0
$686$	−7222.00	−0.401949
$687$	0	0
$688$	−848.000	−0.0469908
$689$	−11220.0	−0.620389
$690$	0	0
$691$	6212.00	0.341991	0.170995	−	0.985272i	$-0.445302\pi$
$691$	6212.00	0.341991	0.170995	+	0.985272i	$0.445302\pi$
$692$	−768.000	−0.0421893
$693$	0	0
$694$	16536.0	0.904464
$695$	0	0
$696$	0	0
$697$	5796.00	0.314977
$698$	−2758.00	−0.149559
$699$	0	0
$700$	0	0
$701$	13224.0	0.712502	0.356251	−	0.934390i	$-0.384055\pi$
$701$	13224.0	0.712502	0.356251	+	0.934390i	$0.384055\pi$
$702$	0	0
$703$	26549.0	1.42434
$704$	−1920.00	−0.102788
$705$	0	0
$706$	−6144.00	−0.327525
$707$	−29394.0	−1.56361
$708$	0	0
$709$	34709.0	1.83854	0.919269	−	0.393629i	$-0.128781\pi$
$709$	34709.0	1.83854	0.919269	+	0.393629i	$0.128781\pi$
$710$	0	0
$711$	0	0
$712$	−6528.00	−0.343606
$713$	−10560.0	−0.554664
$714$	0	0
$715$	0	0
$716$	−4560.00	−0.238010
$717$	0	0
$718$	−14892.0	−0.774045
$719$	−32070.0	−1.66343	−0.831717	−	0.555200i	$-0.812642\pi$
$719$	−32070.0	−1.66343	−0.831717	+	0.555200i	$0.812642\pi$
$720$	0	0
$721$	13915.0	0.718754
$722$	−24924.0	−1.28473
$723$	0	0
$724$	1592.00	0.0817213
$725$	0	0
$726$	0	0
$727$	−125.000	−0.00637688	−0.00318844	−	0.999995i	$-0.501015\pi$
$727$	−125.000	−0.00637688	−0.00318844	+	0.999995i	$0.501015\pi$
$728$	6256.00	0.318493
$729$	0	0
$730$	0	0
$731$	2226.00	0.112629
$732$	0	0
$733$	−32222.0	−1.62367	−0.811833	−	0.583890i	$-0.801530\pi$
$733$	−32222.0	−1.62367	−0.811833	+	0.583890i	$0.801530\pi$
$734$	−12992.0	−0.653329
$735$	0	0
$736$	−6144.00	−0.307705
$737$	13560.0	0.677733
$738$	0	0
$739$	−19240.0	−0.957720	−0.478860	−	0.877891i	$-0.658950\pi$
$739$	−19240.0	−0.957720	−0.478860	+	0.877891i	$0.658950\pi$
$740$	0	0
$741$	0	0
$742$	−15180.0	−0.751045
$743$	30252.0	1.49373	0.746863	−	0.664978i	$-0.231559\pi$
$743$	30252.0	1.49373	0.746863	+	0.664978i	$0.231559\pi$
$744$	0	0
$745$	0	0
$746$	−3266.00	−0.160291
$747$	0	0
$748$	5040.00	0.246365
$749$	−34224.0	−1.66958
$750$	0	0
$751$	−12517.0	−0.608192	−0.304096	−	0.952641i	$-0.598354\pi$
$751$	−12517.0	−0.608192	−0.304096	+	0.952641i	$0.598354\pi$
$752$	5856.00	0.283971
$753$	0	0
$754$	15912.0	0.768542
$755$	0	0
$756$	0	0
$757$	−2201.00	−0.105676	−0.0528380	−	0.998603i	$-0.516827\pi$
$757$	−2201.00	−0.105676	−0.0528380	+	0.998603i	$0.516827\pi$
$758$	−13576.0	−0.650531
$759$	0	0
$760$	0	0
$761$	−8742.00	−0.416422	−0.208211	−	0.978084i	$-0.566764\pi$
$761$	−8742.00	−0.416422	−0.208211	+	0.978084i	$0.566764\pi$
$762$	0	0
$763$	−13639.0	−0.647136
$764$	−13896.0	−0.658036
$765$	0	0
$766$	−13164.0	−0.620933
$767$	13464.0	0.633842
$768$	0	0
$769$	−28618.0	−1.34199	−0.670996	−	0.741461i	$-0.734133\pi$
$769$	−28618.0	−1.34199	−0.670996	+	0.741461i	$0.734133\pi$
$770$	0	0
$771$	0	0
$772$	10852.0	0.505922
$773$	6594.00	0.306817	0.153409	−	0.988163i	$-0.450975\pi$
$773$	6594.00	0.306817	0.153409	+	0.988163i	$0.450975\pi$
$774$	0	0
$775$	0	0
$776$	7240.00	0.334924
$777$	0	0
$778$	5700.00	0.262667
$779$	19182.0	0.882242
$780$	0	0
$781$	6120.00	0.280398
$782$	16128.0	0.737514
$783$	0	0
$784$	2976.00	0.135569
$785$	0	0
$786$	0	0
$787$	−15881.0	−0.719309	−0.359655	−	0.933085i	$-0.617106\pi$
$787$	−15881.0	−0.719309	−0.359655	+	0.933085i	$0.617106\pi$
$788$	18936.0	0.856050
$789$	0	0
$790$	0	0
$791$	−7452.00	−0.334972
$792$	0	0
$793$	782.000	0.0350185
$794$	14902.0	0.666061
$795$	0	0
$796$	20528.0	0.914065
$797$	−26052.0	−1.15785	−0.578927	−	0.815380i	$-0.696528\pi$
$797$	−26052.0	−1.15785	−0.578927	+	0.815380i	$0.696528\pi$
$798$	0	0
$799$	−15372.0	−0.680629
$800$	0	0
$801$	0	0
$802$	28248.0	1.24373
$803$	−20730.0	−0.911016
$804$	0	0
$805$	0	0
$806$	3740.00	0.163444
$807$	0	0
$808$	−10224.0	−0.445147
$809$	−12648.0	−0.549666	−0.274833	−	0.961492i	$-0.588623\pi$
$809$	−12648.0	−0.549666	−0.274833	+	0.961492i	$0.588623\pi$
$810$	0	0
$811$	27179.0	1.17680	0.588399	−	0.808570i	$-0.299758\pi$
$811$	27179.0	1.17680	0.588399	+	0.808570i	$0.299758\pi$
$812$	21528.0	0.930400
$813$	0	0
$814$	−11460.0	−0.493456
$815$	0	0
$816$	0	0
$817$	7367.00	0.315470
$818$	−12748.0	−0.544894
$819$	0	0
$820$	0	0
$821$	−11874.0	−0.504757	−0.252378	−	0.967629i	$-0.581213\pi$
$821$	−11874.0	−0.504757	−0.252378	+	0.967629i	$0.581213\pi$
$822$	0	0
$823$	18448.0	0.781357	0.390679	−	0.920527i	$-0.372240\pi$
$823$	18448.0	0.781357	0.390679	+	0.920527i	$0.372240\pi$
$824$	4840.00	0.204623
$825$	0	0
$826$	18216.0	0.767331
$827$	−3234.00	−0.135982	−0.0679911	−	0.997686i	$-0.521659\pi$
$827$	−3234.00	−0.135982	−0.0679911	+	0.997686i	$0.521659\pi$
$828$	0	0
$829$	−32155.0	−1.34715	−0.673576	−	0.739118i	$-0.735243\pi$
$829$	−32155.0	−1.34715	−0.673576	+	0.739118i	$0.735243\pi$
$830$	0	0
$831$	0	0
$832$	2176.00	0.0906721
$833$	−7812.00	−0.324934
$834$	0	0
$835$	0	0
$836$	16680.0	0.690060
$837$	0	0
$838$	7896.00	0.325493
$839$	21996.0	0.905109	0.452554	−	0.891737i	$-0.350513\pi$
$839$	21996.0	0.905109	0.452554	+	0.891737i	$0.350513\pi$
$840$	0	0
$841$	30367.0	1.24511
$842$	−25258.0	−1.03379
$843$	0	0
$844$	20960.0	0.854826
$845$	0	0
$846$	0	0
$847$	9913.00	0.402143
$848$	−5280.00	−0.213816
$849$	0	0
$850$	0	0
$851$	−36672.0	−1.47720
$852$	0	0
$853$	−3278.00	−0.131579	−0.0657893	−	0.997834i	$-0.520957\pi$
$853$	−3278.00	−0.131579	−0.0657893	+	0.997834i	$0.520957\pi$
$854$	1058.00	0.0423935
$855$	0	0
$856$	−11904.0	−0.475316
$857$	−21228.0	−0.846131	−0.423066	−	0.906099i	$-0.639046\pi$
$857$	−21228.0	−0.846131	−0.423066	+	0.906099i	$0.639046\pi$
$858$	0	0
$859$	−5767.00	−0.229066	−0.114533	−	0.993419i	$-0.536537\pi$
$859$	−5767.00	−0.229066	−0.114533	+	0.993419i	$0.536537\pi$
$860$	0	0
$861$	0	0
$862$	9684.00	0.382643
$863$	−5322.00	−0.209922	−0.104961	−	0.994476i	$-0.533472\pi$
$863$	−5322.00	−0.209922	−0.104961	+	0.994476i	$0.533472\pi$
$864$	0	0
$865$	0	0
$866$	7702.00	0.302222
$867$	0	0
$868$	5060.00	0.197866
$869$	21270.0	0.830305
$870$	0	0
$871$	−15368.0	−0.597847
$872$	−4744.00	−0.184234
$873$	0	0
$874$	53376.0	2.06576
$875$	0	0
$876$	0	0
$877$	42097.0	1.62088	0.810442	−	0.585819i	$-0.199227\pi$
$877$	42097.0	1.62088	0.810442	+	0.585819i	$0.199227\pi$
$878$	14870.0	0.571570
$879$	0	0
$880$	0	0
$881$	17658.0	0.675270	0.337635	−	0.941277i	$-0.390373\pi$
$881$	17658.0	0.675270	0.337635	+	0.941277i	$0.390373\pi$
$882$	0	0
$883$	22297.0	0.849778	0.424889	−	0.905246i	$-0.360313\pi$
$883$	22297.0	0.849778	0.424889	+	0.905246i	$0.360313\pi$
$884$	−5712.00	−0.217325
$885$	0	0
$886$	11520.0	0.436819
$887$	−10542.0	−0.399059	−0.199530	−	0.979892i	$-0.563941\pi$
$887$	−10542.0	−0.399059	−0.199530	+	0.979892i	$0.563941\pi$
$888$	0	0
$889$	44344.0	1.67295
$890$	0	0
$891$	0	0
$892$	18076.0	0.678508
$893$	−50874.0	−1.90642
$894$	0	0
$895$	0	0
$896$	2944.00	0.109768
$897$	0	0
$898$	−4380.00	−0.162764
$899$	12870.0	0.477462
$900$	0	0
$901$	13860.0	0.512479
$902$	−8280.00	−0.305647
$903$	0	0
$904$	−2592.00	−0.0953635
$905$	0	0
$906$	0	0
$907$	−47639.0	−1.74402	−0.872010	−	0.489487i	$-0.837184\pi$
$907$	−47639.0	−1.74402	−0.872010	+	0.489487i	$0.837184\pi$
$908$	20256.0	0.740329
$909$	0	0
$910$	0	0
$911$	10326.0	0.375539	0.187769	−	0.982213i	$-0.439874\pi$
$911$	10326.0	0.375539	0.187769	+	0.982213i	$0.439874\pi$
$912$	0	0
$913$	−32940.0	−1.19404
$914$	14404.0	0.521271
$915$	0	0
$916$	10292.0	0.371242
$917$	63066.0	2.27113
$918$	0	0
$919$	5147.00	0.184748	0.0923742	−	0.995724i	$-0.470554\pi$
$919$	5147.00	0.184748	0.0923742	+	0.995724i	$0.470554\pi$
$920$	0	0
$921$	0	0
$922$	−26952.0	−0.962708
$923$	−6936.00	−0.247347
$924$	0	0
$925$	0	0
$926$	−21686.0	−0.769596
$927$	0	0
$928$	7488.00	0.264877
$929$	47064.0	1.66213	0.831066	−	0.556175i	$-0.187731\pi$
$929$	47064.0	1.66213	0.831066	+	0.556175i	$0.187731\pi$
$930$	0	0
$931$	−25854.0	−0.910130
$932$	16392.0	0.576114
$933$	0	0
$934$	−12216.0	−0.427965
$935$	0	0
$936$	0	0
$937$	15385.0	0.536399	0.268200	−	0.963363i	$-0.413571\pi$
$937$	15385.0	0.536399	0.268200	+	0.963363i	$0.413571\pi$
$938$	−20792.0	−0.723756
$939$	0	0
$940$	0	0
$941$	34914.0	1.20953	0.604763	−	0.796406i	$-0.293268\pi$
$941$	34914.0	1.20953	0.604763	+	0.796406i	$0.293268\pi$
$942$	0	0
$943$	−26496.0	−0.914982
$944$	6336.00	0.218453
$945$	0	0
$946$	−3180.00	−0.109293
$947$	37434.0	1.28452	0.642261	−	0.766486i	$-0.277997\pi$
$947$	37434.0	1.28452	0.642261	+	0.766486i	$0.277997\pi$
$948$	0	0
$949$	23494.0	0.803633
$950$	0	0
$951$	0	0
$952$	−7728.00	−0.263094
$953$	8778.00	0.298371	0.149185	−	0.988809i	$-0.452335\pi$
$953$	8778.00	0.298371	0.149185	+	0.988809i	$0.452335\pi$
$954$	0	0
$955$	0	0
$956$	1224.00	0.0414090
$957$	0	0
$958$	19704.0	0.664517
$959$	−30498.0	−1.02694
$960$	0	0
$961$	−26766.0	−0.898459
$962$	12988.0	0.435291
$963$	0	0
$964$	25928.0	0.866270
$965$	0	0
$966$	0	0
$967$	54061.0	1.79781	0.898906	−	0.438141i	$-0.144363\pi$
$967$	54061.0	1.79781	0.898906	+	0.438141i	$0.144363\pi$
$968$	3448.00	0.114486
$969$	0	0
$970$	0	0
$971$	3084.00	0.101926	0.0509631	−	0.998701i	$-0.483771\pi$
$971$	3084.00	0.101926	0.0509631	+	0.998701i	$0.483771\pi$
$972$	0	0
$973$	−20539.0	−0.676722
$974$	15592.0	0.512936
$975$	0	0
$976$	368.000	0.0120691
$977$	12048.0	0.394524	0.197262	−	0.980351i	$-0.436795\pi$
$977$	12048.0	0.394524	0.197262	+	0.980351i	$0.436795\pi$
$978$	0	0
$979$	−24480.0	−0.799167
$980$	0	0
$981$	0	0
$982$	4908.00	0.159491
$983$	33618.0	1.09079	0.545396	−	0.838179i	$-0.316379\pi$
$983$	33618.0	1.09079	0.545396	+	0.838179i	$0.316379\pi$
$984$	0	0
$985$	0	0
$986$	−19656.0	−0.634863
$987$	0	0
$988$	−18904.0	−0.608721
$989$	−10176.0	−0.327177
$990$	0	0
$991$	25043.0	0.802742	0.401371	−	0.915916i	$-0.368534\pi$
$991$	25043.0	0.802742	0.401371	+	0.915916i	$0.368534\pi$
$992$	1760.00	0.0563307
$993$	0	0
$994$	−9384.00	−0.299439
$995$	0	0
$996$	0	0
$997$	17710.0	0.562569	0.281285	−	0.959624i	$-0.409240\pi$
$997$	17710.0	0.562569	0.281285	+	0.959624i	$0.409240\pi$
$998$	5906.00	0.187326
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.4.a.b.1.1	✓	1
3.2	odd	2		1350.4.a.p.1.1	yes	1
5.2	odd	4		1350.4.c.e.649.1		2
5.3	odd	4		1350.4.c.e.649.2		2
5.4	even	2		1350.4.a.ba.1.1	yes	1
15.2	even	4		1350.4.c.p.649.2		2
15.8	even	4		1350.4.c.p.649.1		2
15.14	odd	2		1350.4.a.m.1.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1350.4.a.b.1.1	✓	1	1.1	even	1	trivial
1350.4.a.m.1.1	yes	1	15.14	odd	2
1350.4.a.p.1.1	yes	1	3.2	odd	2
1350.4.a.ba.1.1	yes	1	5.4	even	2
1350.4.c.e.649.1		2	5.2	odd	4
1350.4.c.e.649.2		2	5.3	odd	4
1350.4.c.p.649.1		2	15.8	even	4
1350.4.c.p.649.2		2	15.2	even	4

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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