LMFDB - Embedded newform 475.3.c.d.151.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,3,Mod(151,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("475.151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	475.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$12.9428125571$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.7600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 19 $ x^4 + 9*x^2 + 19
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			151.3
Root			$1.83901i$ of defining polynomial
Character	$\chi$	$=$	475.151
Dual form			475.3.c.d.151.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.13657i q^{2} -3.67802i q^{3} +2.70820 q^{4} +4.18034 q^{6} +7.62436i q^{8} -4.52786 q^{9} +O(q^{10})$	$q+1.13657i q^{2} -3.67802i q^{3} +2.70820 q^{4} +4.18034 q^{6} +7.62436i q^{8} -4.52786 q^{9} +13.4164 q^{11} -9.96084i q^{12} +20.1266i q^{13} +2.16718 q^{16} -5.14624i q^{18} +19.0000 q^{19} +15.2487i q^{22} +28.0426 q^{24} -22.8754 q^{26} -16.4486i q^{27} +32.9606i q^{32} -49.3459i q^{33} -12.2624 q^{36} -73.3555i q^{37} +21.5949i q^{38} +74.0263 q^{39} +36.3344 q^{44} -7.97096i q^{48} -49.0000 q^{49} +54.5071i q^{52} -69.4725i q^{53} +18.6950 q^{54} -69.8825i q^{57} +120.748 q^{61} -28.7933 q^{64} +56.0851 q^{66} +122.701i q^{67} -34.5221i q^{72} +83.3738 q^{74} +51.4559 q^{76} +84.1362i q^{78} -101.249 q^{81} +102.291i q^{88} +121.230 q^{96} +185.023i q^{97} -55.6920i q^{98} -60.7477 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 16 q^{4} - 28 q^{6} - 36 q^{9}+O(q^{10}) $ 4 * q - 16 * q^4 - 28 * q^6 - 36 * q^9	$ 4 q - 16 q^{4} - 28 q^{6} - 36 q^{9} + 116 q^{16} + 76 q^{19} + 300 q^{24} - 172 q^{26} + 264 q^{36} - 8 q^{39} + 360 q^{44} - 196 q^{49} + 200 q^{54} - 464 q^{64} + 600 q^{66} + 92 q^{74} - 304 q^{76} - 244 q^{81} - 812 q^{96} + 240 q^{99}+O(q^{100}) $ 4 * q - 16 * q^4 - 28 * q^6 - 36 * q^9 + 116 * q^16 + 76 * q^19 + 300 * q^24 - 172 * q^26 + 264 * q^36 - 8 * q^39 + 360 * q^44 - 196 * q^49 + 200 * q^54 - 464 * q^64 + 600 * q^66 + 92 * q^74 - 304 * q^76 - 244 * q^81 - 812 * q^96 + 240 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.13657i	0.568286i	0.958782	+	0.284143i	$0.0917091\pi$
$2$	1.13657i	0.568286i	−0.958782	+	0.284143i	$0.908291\pi$
$3$	− 3.67802i	− 1.22601i	−0.790080	−	0.613004i	$-0.789961\pi$
$3$	− 3.67802i	− 1.22601i	0.790080	−	0.613004i	$-0.210039\pi$
$4$	2.70820	0.677051
$5$	0	0
$5$	0	0
$6$	4.18034	0.696723
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	7.62436i	0.953045i
$9$	−4.52786	−0.503096
$10$	0	0
$11$	13.4164	1.21967	0.609837	−	0.792527i	$-0.291235\pi$
$11$	13.4164	1.21967	0.609837	+	0.792527i	$0.291235\pi$
$12$	− 9.96084i	− 0.830070i
$13$	20.1266i	1.54820i	0.633061	+	0.774102i	$0.281798\pi$
$13$	20.1266i	1.54820i	−0.633061	+	0.774102i	$0.718202\pi$
$14$	0	0
$15$	0	0
$16$	2.16718	0.135449
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	− 5.14624i	− 0.285902i
$19$	19.0000	1.00000
$19$	19.0000	1.00000
$20$	0	0
$21$	0	0
$22$	15.2487i	0.693123i
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	28.0426	1.16844
$25$	0	0
$26$	−22.8754	−0.879823
$27$	− 16.4486i	− 0.609208i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.9606i	1.03002i
$33$	− 49.3459i	− 1.49533i
$34$	0	0
$35$	0	0
$36$	−12.2624	−0.340622
$37$	− 73.3555i	− 1.98258i	−0.131691	−	0.991291i	$-0.542041\pi$
$37$	− 73.3555i	− 1.98258i	0.131691	−	0.991291i	$-0.457959\pi$
$38$	21.5949i	0.568286i
$39$	74.0263	1.89811
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	36.3344	0.825781
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	− 7.97096i	− 0.166062i
$49$	−49.0000	−1.00000
$50$	0	0
$51$	0	0
$52$	54.5071i	1.04821i
$53$	− 69.4725i	− 1.31080i	−0.755281	−	0.655401i	$-0.772500\pi$
$53$	− 69.4725i	− 1.31080i	0.755281	−	0.655401i	$-0.227500\pi$
$54$	18.6950	0.346205
$55$	0	0
$56$	0	0
$57$	− 69.8825i	− 1.22601i
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	120.748	1.97947	0.989735	−	0.142915i	$-0.0456475\pi$
$61$	120.748	1.97947	0.989735	+	0.142915i	$0.0456475\pi$
$62$	0	0
$63$	0	0
$64$	−28.7933	−0.449896
$65$	0	0
$66$	56.0851	0.849775
$67$	122.701i	1.83136i	0.401903	+	0.915682i	$0.368349\pi$
$67$	122.701i	1.83136i	−0.401903	+	0.915682i	$0.631651\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	− 34.5221i	− 0.479473i
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	83.3738	1.12667
$75$	0	0
$76$	51.4559	0.677051
$77$	0	0
$78$	84.1362i	1.07867i
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	−101.249	−1.24999
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	102.291i	1.16240i
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	121.230	1.26281
$97$	185.023i	1.90745i	0.300677	+	0.953726i	$0.402787\pi$
$97$	185.023i	1.90745i	−0.300677	+	0.953726i	$0.597213\pi$
$98$	− 55.6920i	− 0.568286i
$99$	−60.7477	−0.613613
$100$	0	0
$101$	−201.246	−1.99254	−0.996268	−	0.0863148i	$-0.972491\pi$
$101$	−201.246	−1.99254	−0.996268	+	0.0863148i	$0.972491\pi$
$102$	0	0
$103$	81.1215i	0.787587i	0.919199	+	0.393794i	$0.128838\pi$
$103$	81.1215i	0.787587i	−0.919199	+	0.393794i	$0.871162\pi$
$104$	−153.453	−1.47551
$105$	0	0
$106$	78.9605	0.744911
$107$	− 74.6821i	− 0.697964i	−0.937129	−	0.348982i	$-0.886527\pi$
$107$	− 74.6821i	− 0.697964i	0.937129	−	0.348982i	$-0.113473\pi$
$108$	− 44.5462i	− 0.412465i
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−269.803	−2.43066
$112$	0	0
$113$	− 40.8682i	− 0.361666i	−0.983514	−	0.180833i	$-0.942121\pi$
$113$	− 40.8682i	− 0.361666i	0.983514	−	0.180833i	$-0.0578793\pi$
$114$	79.4265	0.696723
$115$	0	0
$116$	0	0
$117$	− 91.1307i	− 0.778895i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	59.0000	0.487603
$122$	137.238i	1.12491i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 134.350i	− 1.05788i	−0.848660	−	0.528938i	$-0.822590\pi$
$127$	− 134.350i	− 1.05788i	0.848660	−	0.528938i	$-0.177410\pi$
$128$	99.1166i	0.774349i
$129$	0	0
$130$	0	0
$131$	−118.000	−0.900763	−0.450382	−	0.892836i	$-0.648712\pi$
$131$	−118.000	−0.900763	−0.450382	+	0.892836i	$0.648712\pi$
$132$	− 133.639i	− 1.01241i
$133$	0	0
$134$	−139.459	−1.04074
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	201.246	1.44781	0.723907	−	0.689898i	$-0.242344\pi$
$139$	201.246	1.44781	0.723907	+	0.689898i	$0.242344\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	270.027i	1.88830i
$144$	−9.81272	−0.0681439
$145$	0	0
$146$	0	0
$147$	180.223i	1.22601i
$148$	− 198.662i	− 1.34231i
$149$	−228.079	−1.53073	−0.765366	−	0.643596i	$-0.777442\pi$
$149$	−228.079	−1.53073	−0.765366	+	0.643596i	$0.777442\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	144.863i	0.953045i
$153$	0	0
$154$	0	0
$155$	0	0
$156$	200.478	1.28512
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	−255.522	−1.60705
$160$	0	0
$161$	0	0
$162$	− 115.077i	− 0.710352i
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 331.734i	− 1.98643i	−0.116291	−	0.993215i	$-0.537100\pi$
$167$	− 331.734i	− 1.98643i	0.116291	−	0.993215i	$-0.462900\pi$
$168$	0	0
$169$	−236.082	−1.39694
$170$	0	0
$171$	−86.0294	−0.503096
$172$	0	0
$173$	− 299.247i	− 1.72975i	−0.501988	−	0.864874i	$-0.667398\pi$
$173$	− 299.247i	− 1.72975i	0.501988	−	0.864874i	$-0.332602\pi$
$174$	0	0
$175$	0	0
$176$	29.0758	0.165204
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	− 444.113i	− 2.42685i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−302.000	−1.58115	−0.790576	−	0.612364i	$-0.790219\pi$
$191$	−302.000	−1.58115	−0.790576	+	0.612364i	$0.790219\pi$
$192$	105.903i	0.551576i
$193$	66.9161i	0.346716i	0.984859	+	0.173358i	$0.0554617\pi$
$193$	66.9161i	0.346716i	−0.984859	+	0.173358i	$0.944538\pi$
$194$	−210.292	−1.08398
$195$	0	0
$196$	−132.702	−0.677051
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	− 69.0441i	− 0.348708i
$199$	−322.000	−1.61809	−0.809045	−	0.587746i	$-0.800015\pi$
$199$	−322.000	−1.61809	−0.809045	+	0.587746i	$0.800015\pi$
$200$	0	0
$201$	451.299	2.24527
$202$	− 228.731i	− 1.13233i
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−92.2004	−0.447575
$207$	0	0
$208$	43.6182i	0.209703i
$209$	254.912	1.21967
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	− 188.146i	− 0.887480i
$213$	0	0
$214$	84.8816	0.396643
$215$	0	0
$216$	125.410	0.580603
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	− 306.651i	− 1.38131i
$223$	386.289i	1.73224i	0.499837	+	0.866120i	$0.333393\pi$
$223$	386.289i	1.73224i	−0.499837	+	0.866120i	$0.666607\pi$
$224$	0	0
$225$	0	0
$226$	46.4497	0.205529
$227$	442.075i	1.94747i	0.227694	+	0.973733i	$0.426881\pi$
$227$	442.075i	1.94747i	−0.227694	+	0.973733i	$0.573119\pi$
$228$	− 189.256i	− 0.830070i
$229$	415.909	1.81619	0.908097	−	0.418759i	$-0.137535\pi$
$229$	415.909	1.81619	0.908097	+	0.418759i	$0.137535\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	103.577	0.442635
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−98.0000	−0.410042	−0.205021	−	0.978758i	$-0.565726\pi$
$239$	−98.0000	−0.410042	−0.205021	+	0.978758i	$0.565726\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	67.0578i	0.277098i
$243$	224.359i	0.923290i
$244$	327.009	1.34020
$245$	0	0
$246$	0	0
$247$	382.406i	1.54820i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−182.000	−0.725100	−0.362550	−	0.931964i	$-0.618094\pi$
$251$	−182.000	−0.725100	−0.362550	+	0.931964i	$0.618094\pi$
$252$	0	0
$253$	0	0
$254$	152.699	0.601177
$255$	0	0
$256$	−227.827	−0.889948
$257$	504.396i	1.96263i	0.192407	+	0.981315i	$0.438371\pi$
$257$	504.396i	1.96263i	−0.192407	+	0.981315i	$0.561629\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	− 134.116i	− 0.511891i
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	376.231	1.42512
$265$	0	0
$266$	0	0
$267$	0	0
$268$	332.300i	1.23993i
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−40.2492	−0.148521	−0.0742606	−	0.997239i	$-0.523660\pi$
$271$	−40.2492	−0.148521	−0.0742606	+	0.997239i	$0.523660\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	228.731i	0.822772i
$279$	0	0
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	−306.906	−1.07310
$287$	0	0
$288$	− 149.241i	− 0.518198i
$289$	−289.000	−1.00000
$290$	0	0
$291$	680.519	2.33855
$292$	0	0
$293$	5.92126i	0.0202091i	0.999949	+	0.0101045i	$0.00321643\pi$
$293$	5.92126i	0.0202091i	−0.999949	+	0.0101045i	$0.996784\pi$
$294$	−204.837	−0.696723
$295$	0	0
$296$	559.289	1.88949
$297$	− 220.681i	− 0.743035i
$298$	− 259.228i	− 0.869893i
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	740.188i	2.44287i
$304$	41.1765	0.135449
$305$	0	0
$306$	0	0
$307$	47.3076i	0.154096i	0.997027	+	0.0770482i	$0.0245495\pi$
$307$	47.3076i	0.154096i	−0.997027	+	0.0770482i	$0.975450\pi$
$308$	0	0
$309$	298.367	0.965589
$310$	0	0
$311$	−362.243	−1.16477	−0.582384	−	0.812914i	$-0.697880\pi$
$311$	−362.243	−1.16477	−0.582384	+	0.812914i	$0.697880\pi$
$312$	564.403i	1.80898i
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 516.045i	− 1.62790i	−0.580933	−	0.813951i	$-0.697312\pi$
$317$	− 516.045i	− 1.62790i	0.580933	−	0.813951i	$-0.302688\pi$
$318$	− 290.419i	− 0.913267i
$319$	0	0
$320$	0	0
$321$	−274.683	−0.855709
$322$	0	0
$323$	0	0
$324$	−274.204	−0.846307
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	332.144i	0.997429i
$334$	377.039	1.12886
$335$	0	0
$336$	0	0
$337$	− 60.2832i	− 0.178882i	−0.995992	−	0.0894409i	$-0.971492\pi$
$337$	− 60.2832i	− 0.178882i	0.995992	−	0.0894409i	$-0.0285080\pi$
$338$	− 268.324i	− 0.793859i
$339$	−150.314	−0.443405
$340$	0	0
$341$	0	0
$342$	− 97.7786i	− 0.285902i
$343$	0	0
$344$	0	0
$345$	0	0
$346$	340.115	0.982992
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	518.000	1.48424	0.742120	−	0.670267i	$-0.233820\pi$
$349$	518.000	1.48424	0.742120	+	0.670267i	$0.233820\pi$
$350$	0	0
$351$	331.056	0.943179
$352$	442.213i	1.25629i
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−67.0820	−0.186858	−0.0934290	−	0.995626i	$-0.529783\pi$
$359$	−67.0820	−0.186858	−0.0934290	+	0.995626i	$0.529783\pi$
$360$	0	0
$361$	361.000	1.00000
$362$	0	0
$363$	− 217.003i	− 0.597806i
$364$	0	0
$365$	0	0
$366$	504.766	1.37914
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 708.219i	− 1.89871i	−0.314205	−	0.949355i	$-0.601738\pi$
$373$	− 708.219i	− 1.89871i	0.314205	−	0.949355i	$-0.398262\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−494.144	−1.29697
$382$	− 343.245i	− 0.898547i
$383$	− 755.008i	− 1.97130i	−0.168798	−	0.985651i	$-0.553988\pi$
$383$	− 755.008i	− 1.97130i	0.168798	−	0.985651i	$-0.446012\pi$
$384$	364.553	0.949358
$385$	0	0
$386$	−76.0550	−0.197034
$387$	0	0
$388$	501.080i	1.29144i
$389$	742.000	1.90746	0.953728	−	0.300672i	$-0.0972110\pi$
$389$	742.000	1.90746	0.953728	+	0.300672i	$0.0972110\pi$
$390$	0	0
$391$	0	0
$392$	− 373.594i	− 0.953045i
$393$	434.007i	1.10434i
$394$	0	0
$395$	0	0
$396$	−164.517	−0.415447
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	− 365.976i	− 0.919538i
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	512.934i	1.27595i
$403$	0	0
$404$	−545.016	−1.34905
$405$	0	0
$406$	0	0
$407$	− 984.168i	− 2.41810i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	219.694i	0.533237i
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−663.386	−1.59468
$417$	− 740.188i	− 1.77503i
$418$	289.726i	0.693123i
$419$	−458.000	−1.09308	−0.546539	−	0.837433i	$-0.684055\pi$
$419$	−458.000	−1.09308	−0.546539	+	0.837433i	$0.684055\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	529.683	1.24925
$425$	0	0
$426$	0	0
$427$	0	0
$428$	− 202.254i	− 0.472557i
$429$	993.167	2.31508
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	− 35.6472i	− 0.0825167i
$433$	− 679.615i	− 1.56955i	−0.619781	−	0.784774i	$-0.712779\pi$
$433$	− 679.615i	− 1.56955i	0.619781	−	0.784774i	$-0.287221\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	221.865	0.503096
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	−730.683	−1.64568
$445$	0	0
$446$	−439.046	−0.984407
$447$	838.880i	1.87669i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	− 110.679i	− 0.244866i
$453$	0	0
$454$	−502.450	−1.10672
$455$	0	0
$456$	532.809	1.16844
$457$	0	0		−	1.00000i	$-0.5\pi$
$457$	0	0			1.00000i	$0.5\pi$
$458$	472.710i	1.03212i
$459$	0	0
$460$	0	0
$461$	−598.000	−1.29718	−0.648590	−	0.761138i	$-0.724641\pi$
$461$	−598.000	−1.29718	−0.648590	+	0.761138i	$0.724641\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	− 246.801i	− 0.527352i
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	314.562i	0.659459i
$478$	− 111.384i	− 0.233021i
$479$	898.899	1.87662	0.938308	−	0.345800i	$-0.112392\pi$
$479$	898.899	1.87662	0.938308	+	0.345800i	$0.112392\pi$
$480$	0	0
$481$	1476.40	3.06944
$482$	0	0
$483$	0	0
$484$	159.784	0.330132
$485$	0	0
$486$	−255.001	−0.524693
$487$	− 971.807i	− 1.99550i	−0.0670685	−	0.997748i	$-0.521365\pi$
$487$	− 971.807i	− 1.99550i	0.0670685	−	0.997748i	$-0.478635\pi$
$488$	920.623i	1.88652i
$489$	0	0
$490$	0	0
$491$	298.000	0.606925	0.303462	−	0.952843i	$-0.401857\pi$
$491$	298.000	0.606925	0.303462	+	0.952843i	$0.401857\pi$
$492$	0	0
$493$	0	0
$494$	−434.632	−0.879823
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−979.398	−1.96272	−0.981360	−	0.192176i	$-0.938446\pi$
$499$	−979.398	−1.96272	−0.981360	+	0.192176i	$0.938446\pi$
$500$	0	0
$501$	−1220.13	−2.43538
$502$	− 206.856i	− 0.412064i
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	868.316i	1.71265i
$508$	− 363.848i	− 0.716237i
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	137.525i	0.268604i
$513$	− 312.524i	− 0.609208i
$514$	−573.282	−1.11534
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−1100.64	−2.12069
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	888.647i	1.69913i	0.527481	+	0.849567i	$0.323137\pi$
$523$	888.647i	1.69913i	−0.527481	+	0.849567i	$0.676863\pi$
$524$	−319.568	−0.609863
$525$	0	0
$526$	0	0
$527$	0	0
$528$	− 106.942i	− 0.202541i
$529$	−529.000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	−935.519	−1.74537
$537$	0	0
$538$	0	0
$539$	−657.404	−1.21967
$540$	0	0
$541$	335.410	0.619982	0.309991	−	0.950740i	$-0.399674\pi$
$541$	335.410	0.619982	0.309991	+	0.950740i	$0.399674\pi$
$542$	− 45.7461i	− 0.0844025i
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 119.951i	− 0.219290i	−0.993971	−	0.109645i	$-0.965029\pi$
$547$	− 119.951i	− 0.219290i	0.993971	−	0.109645i	$-0.0349713\pi$
$548$	0	0
$549$	−546.729	−0.995863
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	545.016	0.980244
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1086.03i	1.92901i	0.264071	+	0.964503i	$0.414935\pi$
$563$	1086.03i	1.92901i	−0.264071	+	0.964503i	$0.585065\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−415.909	−0.728386	−0.364193	−	0.931323i	$-0.618655\pi$
$571$	−415.909	−0.728386	−0.364193	+	0.931323i	$0.618655\pi$
$572$	731.289i	1.27848i
$573$	1110.76i	1.93851i
$574$	0	0
$575$	0	0
$576$	130.372	0.226341
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	− 328.469i	− 0.568286i
$579$	246.119	0.425076
$580$	0	0
$581$	0	0
$582$	773.458i	1.32897i
$583$	− 932.072i	− 1.59875i
$584$	0	0
$585$	0	0
$586$	−6.72994	−0.0114845
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	488.081i	0.830070i
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 158.975i	− 0.268539i
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	250.820	0.422257
$595$	0	0
$596$	−617.684	−1.03638
$597$	1184.32i	1.98379i
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	− 555.575i	− 0.921352i
$604$	0	0
$605$	0	0
$606$	−841.277	−1.38825
$607$	653.857i	1.07719i	0.842563	+	0.538597i	$0.181046\pi$
$607$	653.857i	1.07719i	−0.842563	+	0.538597i	$0.818954\pi$
$608$	626.251i	1.03002i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−53.7685	−0.0875709
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	339.115i	0.548731i
$619$	−13.4164	−0.0216743	−0.0108372	−	0.999941i	$-0.503450\pi$
$619$	−13.4164	−0.0216743	−0.0108372	+	0.999941i	$0.503450\pi$
$620$	0	0
$621$	0	0
$622$	− 411.715i	− 0.661922i
$623$	0	0
$624$	160.429	0.257097
$625$	0	0
$626$	0	0
$627$	− 937.572i	− 1.49533i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	1140.39	1.80728	0.903641	−	0.428291i	$-0.140884\pi$
$631$	1140.39	1.80728	0.903641	+	0.428291i	$0.140884\pi$
$632$	0	0
$633$	0	0
$634$	586.522	0.925114
$635$	0	0
$636$	−692.005	−1.08806
$637$	− 986.206i	− 1.54820i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	− 312.197i	− 0.486288i
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	− 771.960i	− 1.19130i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	−377.505	−0.566825
$667$	0	0
$668$	− 898.403i	− 1.34491i
$669$	1420.78	2.12374
$670$	0	0
$671$	1620.00	2.41431
$672$	0	0
$673$	− 604.221i	− 0.897802i	−0.893581	−	0.448901i	$-0.851816\pi$
$673$	− 604.221i	− 0.897802i	0.893581	−	0.448901i	$-0.148184\pi$
$674$	68.5162	0.101656
$675$	0	0
$676$	−639.358	−0.945796
$677$	1204.14i	1.77864i	0.457288	+	0.889319i	$0.348821\pi$
$677$	1204.14i	1.77864i	−0.457288	+	0.889319i	$0.651179\pi$
$678$	− 170.843i	− 0.251981i
$679$	0	0
$680$	0	0
$681$	1625.96	2.38761
$682$	0	0
$683$	− 539.633i	− 0.790093i	−0.918661	−	0.395046i	$-0.870729\pi$
$683$	− 539.633i	− 0.790093i	0.918661	−	0.395046i	$-0.129271\pi$
$684$	−232.985	−0.340622
$685$	0	0
$686$	0	0
$687$	− 1529.72i	− 2.22667i
$688$	0	0
$689$	1398.25	2.02939
$690$	0	0
$691$	−1381.89	−1.99984	−0.999920	−	0.0126162i	$-0.995984\pi$
$691$	−1381.89	−1.99984	−0.999920	+	0.0126162i	$0.995984\pi$
$692$	− 810.421i	− 1.17113i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	588.744i	0.843473i
$699$	0	0
$700$	0	0
$701$	−952.565	−1.35887	−0.679433	−	0.733738i	$-0.737774\pi$
$701$	−952.565	−1.35887	−0.679433	+	0.733738i	$0.737774\pi$
$702$	376.269i	0.535995i
$703$	− 1393.75i	− 1.98258i
$704$	−386.303	−0.548726
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−202.000	−0.284908	−0.142454	−	0.989801i	$-0.545499\pi$
$709$	−202.000	−0.284908	−0.142454	+	0.989801i	$0.545499\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	360.446i	0.502715i
$718$	− 76.2436i	− 0.106189i
$719$	−1247.73	−1.73536	−0.867681	−	0.497121i	$-0.834391\pi$
$719$	−1247.73	−1.73536	−0.867681	+	0.497121i	$0.834391\pi$
$720$	0	0
$721$	0	0
$722$	410.303i	0.568286i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	246.640	0.339725
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	−86.0433	−0.118029
$730$	0	0
$731$	0	0
$732$	− 1202.75i	− 1.64310i
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1646.21i	2.23367i
$738$	0	0
$739$	−1402.00	−1.89716	−0.948579	−	0.316540i	$-0.897479\pi$
$739$	−1402.00	−1.89716	−0.948579	+	0.316540i	$0.897479\pi$
$740$	0	0
$741$	1406.50	1.89811
$742$	0	0
$743$	− 1483.35i	− 1.99644i	−0.0596489	−	0.998219i	$-0.518998\pi$
$743$	− 1483.35i	− 1.99644i	0.0596489	−	0.998219i	$-0.481002\pi$
$744$	0	0
$745$	0	0
$746$	804.942	1.07901
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	669.400i	0.888978i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1435.56	−1.88641	−0.943203	−	0.332216i	$-0.892204\pi$
$761$	−1435.56	−1.88641	−0.943203	+	0.332216i	$0.892204\pi$
$762$	− 561.630i	− 0.737048i
$763$	0	0
$764$	−817.878	−1.07052
$765$	0	0
$766$	858.122	1.12026
$767$	0	0
$768$	837.952i	1.09108i
$769$	684.237	0.889775	0.444887	−	0.895587i	$-0.353244\pi$
$769$	684.237	0.889775	0.444887	+	0.895587i	$0.353244\pi$
$770$	0	0
$771$	1855.18	2.40620
$772$	181.223i	0.234744i
$773$	705.469i	0.912638i	0.889816	+	0.456319i	$0.150832\pi$
$773$	705.469i	0.912638i	−0.889816	+	0.456319i	$0.849168\pi$
$774$	0	0
$775$	0	0
$776$	−1410.68	−1.81789
$777$	0	0
$778$	843.336i	1.08398i
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−106.192	−0.135449
$785$	0	0
$786$	−493.280	−0.627583
$787$	913.562i	1.16082i	0.814326	+	0.580408i	$0.197107\pi$
$787$	913.562i	1.16082i	−0.814326	+	0.580408i	$0.802893\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	− 463.162i	− 0.584800i
$793$	2430.25i	3.06462i
$794$	0	0
$795$	0	0
$796$	−872.042	−1.09553
$797$	− 758.698i	− 0.951942i	−0.879461	−	0.475971i	$-0.842097\pi$
$797$	− 758.698i	− 0.951942i	0.879461	−	0.475971i	$-0.157903\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	1222.21	1.52016
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 1534.37i	− 1.89898i
$809$	1118.00	1.38195	0.690977	−	0.722877i	$-0.257181\pi$
$809$	1118.00	1.38195	0.690977	+	0.722877i	$0.257181\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	148.038i	0.182088i
$814$	1118.58	1.37417
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	122.000	0.148599	0.0742996	−	0.997236i	$-0.476328\pi$
$821$	122.000	0.148599	0.0742996	+	0.997236i	$0.476328\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	−618.499	−0.750606
$825$	0	0
$826$	0	0
$827$	− 1501.64i	− 1.81576i	−0.419227	−	0.907881i	$-0.637699\pi$
$827$	− 1501.64i	− 1.81576i	0.419227	−	0.907881i	$-0.362301\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	− 579.514i	− 0.696531i
$833$	0	0
$834$	841.277	1.00873
$835$	0	0
$836$	690.353	0.825781
$837$	0	0
$838$	− 520.550i	− 0.621181i
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	− 150.560i	− 0.177547i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	569.403	0.665191
$857$	− 1609.23i	− 1.87774i	−0.344266	−	0.938872i	$-0.611872\pi$
$857$	− 1609.23i	− 1.87774i	0.344266	−	0.938872i	$-0.388128\pi$
$858$	1128.81i	1.31562i
$859$	1702.00	1.98137	0.990687	−	0.136160i	$-0.0434762\pi$
$859$	1702.00	1.98137	0.990687	+	0.136160i	$0.0434762\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	217.704i	0.252264i	0.992013	+	0.126132i	$0.0402563\pi$
$863$	217.704i	0.252264i	−0.992013	+	0.126132i	$0.959744\pi$
$864$	542.156	0.627496
$865$	0	0
$866$	772.431	0.891953
$867$	1062.95i	1.22601i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−2469.57	−2.83532
$872$	0	0
$873$	− 837.758i	− 0.959631i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 44.5577i	− 0.0508069i	−0.999677	−	0.0254035i	$-0.991913\pi$
$877$	− 44.5577i	− 0.0508069i	0.999677	−	0.0254035i	$-0.00808704\pi$
$878$	0	0
$879$	21.7786	0.0247765
$880$	0	0
$881$	1677.05	1.90358	0.951788	−	0.306756i	$-0.0992434\pi$
$881$	1677.05	1.90358	0.951788	+	0.306756i	$0.0992434\pi$
$882$	252.166i	0.285902i
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1489.99i	1.67980i	0.542737	+	0.839902i	$0.317388\pi$
$887$	1489.99i	1.67980i	−0.542737	+	0.839902i	$0.682612\pi$
$888$	− 2057.08i	− 2.31653i
$889$	0	0
$890$	0	0
$891$	−1358.40	−1.52458
$892$	1046.15i	1.17281i
$893$	0	0
$894$	−953.447	−1.06650
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	311.594	0.344683
$905$	0	0
$906$	0	0
$907$	1522.18i	1.67826i	0.543929	+	0.839131i	$0.316936\pi$
$907$	1522.18i	1.67826i	−0.543929	+	0.839131i	$0.683064\pi$
$908$	1197.23i	1.31853i
$909$	911.215	1.00244
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	− 151.448i	− 0.166062i
$913$	0	0
$914$	0	0
$915$	0	0
$916$	1126.37	1.22966
$917$	0	0
$918$	0	0
$919$	1582.00	1.72144	0.860718	−	0.509082i	$-0.170015\pi$
$919$	1582.00	1.72144	0.860718	+	0.509082i	$0.170015\pi$
$920$	0	0
$921$	173.999	0.188923
$922$	− 679.670i	− 0.737169i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 367.307i	− 0.396232i
$928$	0	0
$929$	878.000	0.945102	0.472551	−	0.881303i	$-0.343333\pi$
$929$	878.000	0.945102	0.472551	+	0.881303i	$0.343333\pi$
$930$	0	0
$931$	−931.000	−1.00000
$932$	0	0
$933$	1332.34i	1.42802i
$934$	0	0
$935$	0	0
$936$	694.813	0.742322
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−1898.03	−1.99582
$952$	0	0
$953$	− 356.649i	− 0.374238i	−0.982337	−	0.187119i	$-0.940085\pi$
$953$	− 356.649i	− 0.374238i	0.982337	−	0.187119i	$-0.0599150\pi$
$954$	−357.523	−0.374762
$955$	0	0
$956$	−265.404	−0.277619
$957$	0	0
$958$	1021.66i	1.06645i
$959$	0	0
$960$	0	0
$961$	961.000	1.00000
$962$	1678.04i	1.74432i
$963$	338.150i	0.351143i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	449.837i	0.464708i
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	607.611i	0.625114i
$973$	0	0
$974$	1104.53	1.13401
$975$	0	0
$976$	261.682	0.268117
$977$	− 1000.60i	− 1.02416i	−0.858938	−	0.512080i	$-0.828875\pi$
$977$	− 1000.60i	− 1.02416i	0.858938	−	0.512080i	$-0.171125\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	338.698i	0.344907i
$983$	− 12.0703i	− 0.0122791i	−0.999981	−	0.00613954i	$-0.998046\pi$
$983$	− 12.0703i	− 0.0122791i	0.999981	−	0.00613954i	$-0.00195429\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	1035.63i	1.04821i
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	− 1113.16i	− 1.11539i
$999$	−1206.60	−1.20781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.3.c.d.151.3		4
5.2	odd	4		95.3.d.c.94.2	✓	4
5.3	odd	4		95.3.d.c.94.3	yes	4
5.4	even	2	inner	475.3.c.d.151.2		4
15.2	even	4		855.3.g.d.379.3		4
15.8	even	4		855.3.g.d.379.2		4
19.18	odd	2	inner	475.3.c.d.151.2		4
95.18	even	4		95.3.d.c.94.2	✓	4
95.37	even	4		95.3.d.c.94.3	yes	4
95.94	odd	2	CM	475.3.c.d.151.3		4
285.113	odd	4		855.3.g.d.379.3		4
285.227	odd	4		855.3.g.d.379.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.d.c.94.2	✓	4	5.2	odd	4
95.3.d.c.94.2	✓	4	95.18	even	4
95.3.d.c.94.3	yes	4	5.3	odd	4
95.3.d.c.94.3	yes	4	95.37	even	4
475.3.c.d.151.2		4	5.4	even	2	inner
475.3.c.d.151.2		4	19.18	odd	2	inner
475.3.c.d.151.3		4	1.1	even	1	trivial
475.3.c.d.151.3		4	95.94	odd	2	CM
855.3.g.d.379.2		4	15.8	even	4
855.3.g.d.379.2		4	285.227	odd	4
855.3.g.d.379.3		4	15.2	even	4
855.3.g.d.379.3		4	285.113	odd	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:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	475.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.13657i q^{2} -3.67802i q^{3} +2.70820 q^{4} +4.18034 q^{6} +7.62436i q^{8} -4.52786 q^{9} +O(q^{10})\)	\(q+1.13657i q^{2} -3.67802i q^{3} +2.70820 q^{4} +4.18034 q^{6} +7.62436i q^{8} -4.52786 q^{9} +13.4164 q^{11} -9.96084i q^{12} +20.1266i q^{13} +2.16718 q^{16} -5.14624i q^{18} +19.0000 q^{19} +15.2487i q^{22} +28.0426 q^{24} -22.8754 q^{26} -16.4486i q^{27} +32.9606i q^{32} -49.3459i q^{33} -12.2624 q^{36} -73.3555i q^{37} +21.5949i q^{38} +74.0263 q^{39} +36.3344 q^{44} -7.97096i q^{48} -49.0000 q^{49} +54.5071i q^{52} -69.4725i q^{53} +18.6950 q^{54} -69.8825i q^{57} +120.748 q^{61} -28.7933 q^{64} +56.0851 q^{66} +122.701i q^{67} -34.5221i q^{72} +83.3738 q^{74} +51.4559 q^{76} +84.1362i q^{78} -101.249 q^{81} +102.291i q^{88} +121.230 q^{96} +185.023i q^{97} -55.6920i q^{98} -60.7477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 16 q^{4} - 28 q^{6} - 36 q^{9}+O(q^{10}) \) 4 * q - 16 * q^4 - 28 * q^6 - 36 * q^9	\( 4 q - 16 q^{4} - 28 q^{6} - 36 q^{9} + 116 q^{16} + 76 q^{19} + 300 q^{24} - 172 q^{26} + 264 q^{36} - 8 q^{39} + 360 q^{44} - 196 q^{49} + 200 q^{54} - 464 q^{64} + 600 q^{66} + 92 q^{74} - 304 q^{76} - 244 q^{81} - 812 q^{96} + 240 q^{99}+O(q^{100}) \) 4 * q - 16 * q^4 - 28 * q^6 - 36 * q^9 + 116 * q^16 + 76 * q^19 + 300 * q^24 - 172 * q^26 + 264 * q^36 - 8 * q^39 + 360 * q^44 - 196 * q^49 + 200 * q^54 - 464 * q^64 + 600 * q^66 + 92 * q^74 - 304 * q^76 - 244 * q^81 - 812 * q^96 + 240 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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