LMFDB - Embedded newform 3150.3.c.f.449.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.c (of order $2$, degree $1$, not 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:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.9671731157401600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 $ x^16 + 7x^12 + 753x^8 + 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.8
Root			$2.08559 - 0.941471i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.f.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +14.1001i q^{11} -1.78360i q^{13} -3.74166i q^{14} +4.00000 q^{16} +32.2729 q^{17} -9.89548 q^{19} -19.9406i q^{22} +9.58126 q^{23} +2.52238i q^{26} +5.29150i q^{28} -40.5881i q^{29} +48.5626 q^{31} -5.65685 q^{32} -45.6408 q^{34} +62.8976i q^{37} +13.9943 q^{38} -65.7349i q^{41} -50.8420i q^{43} +28.2003i q^{44} -13.5500 q^{46} -65.3405 q^{47} -7.00000 q^{49} -3.56719i q^{52} +64.9575 q^{53} -7.48331i q^{56} +57.4003i q^{58} -111.642i q^{59} +2.69105 q^{61} -68.6778 q^{62} +8.00000 q^{64} +72.1824i q^{67} +64.5458 q^{68} -40.6525i q^{71} +63.6965i q^{73} -88.9506i q^{74} -19.7910 q^{76} -37.3055 q^{77} +82.3035 q^{79} +92.9631i q^{82} -99.9021 q^{83} +71.9014i q^{86} -39.8812i q^{88} +88.5703i q^{89} +4.71895 q^{91} +19.1625 q^{92} +92.4054 q^{94} +117.848i q^{97} +9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) $ 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	−2.82843	−0.353553
$9$	0	0
$10$	0	0
$11$	14.1001i	1.28183i	0.767611	+	0.640916i	$0.221445\pi$
$11$	14.1001i	1.28183i	−0.767611	+	0.640916i	$0.778555\pi$
$12$	0	0
$13$	− 1.78360i	− 0.137200i	−0.997644	−	0.0685998i	$-0.978147\pi$
$13$	− 1.78360i	− 0.137200i	0.997644	−	0.0685998i	$-0.0218532\pi$
$14$	− 3.74166i	− 0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	32.2729	1.89841	0.949203	−	0.314665i	$-0.101892\pi$
$17$	32.2729	1.89841	0.949203	+	0.314665i	$0.101892\pi$
$18$	0	0
$19$	−9.89548	−0.520815	−0.260407	−	0.965499i	$-0.583857\pi$
$19$	−9.89548	−0.520815	−0.260407	+	0.965499i	$0.583857\pi$
$20$	0	0
$21$	0	0
$22$	− 19.9406i	− 0.906392i
$23$	9.58126	0.416577	0.208288	−	0.978067i	$-0.433211\pi$
$23$	9.58126	0.416577	0.208288	+	0.978067i	$0.433211\pi$
$24$	0	0
$25$	0	0
$26$	2.52238i	0.0970148i
$27$	0	0
$28$	5.29150i	0.188982i
$29$	− 40.5881i	− 1.39959i	−0.714343	−	0.699795i	$-0.753274\pi$
$29$	− 40.5881i	− 1.39959i	0.714343	−	0.699795i	$-0.246726\pi$
$30$	0	0
$31$	48.5626	1.56653	0.783267	−	0.621685i	$-0.213552\pi$
$31$	48.5626	1.56653	0.783267	+	0.621685i	$0.213552\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	−45.6408	−1.34238
$35$	0	0
$36$	0	0
$37$	62.8976i	1.69993i	0.526835	+	0.849967i	$0.323378\pi$
$37$	62.8976i	1.69993i	−0.526835	+	0.849967i	$0.676622\pi$
$38$	13.9943	0.368272
$39$	0	0
$40$	0	0
$41$	− 65.7349i	− 1.60329i	−0.597801	−	0.801645i	$-0.703959\pi$
$41$	− 65.7349i	− 1.60329i	0.597801	−	0.801645i	$-0.296041\pi$
$42$	0	0
$43$	− 50.8420i	− 1.18237i	−0.806535	−	0.591186i	$-0.798660\pi$
$43$	− 50.8420i	− 1.18237i	0.806535	−	0.591186i	$-0.201340\pi$
$44$	28.2003i	0.640916i
$45$	0	0
$46$	−13.5500	−0.294564
$47$	−65.3405	−1.39022	−0.695111	−	0.718902i	$-0.744645\pi$
$47$	−65.3405	−1.39022	−0.695111	+	0.718902i	$0.744645\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 3.56719i	− 0.0685998i
$53$	64.9575	1.22561	0.612807	−	0.790233i	$-0.290040\pi$
$53$	64.9575	1.22561	0.612807	+	0.790233i	$0.290040\pi$
$54$	0	0
$55$	0	0
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	57.4003i	0.989660i
$59$	− 111.642i	− 1.89223i	−0.323824	−	0.946117i	$-0.604968\pi$
$59$	− 111.642i	− 1.89223i	0.323824	−	0.946117i	$-0.395032\pi$
$60$	0	0
$61$	2.69105	0.0441156	0.0220578	−	0.999757i	$-0.492978\pi$
$61$	2.69105	0.0441156	0.0220578	+	0.999757i	$0.492978\pi$
$62$	−68.6778	−1.10771
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	72.1824i	1.07735i	0.842514	+	0.538674i	$0.181075\pi$
$67$	72.1824i	1.07735i	−0.842514	+	0.538674i	$0.818925\pi$
$68$	64.5458	0.949203
$69$	0	0
$70$	0	0
$71$	− 40.6525i	− 0.572570i	−0.958144	−	0.286285i	$-0.907580\pi$
$71$	− 40.6525i	− 0.572570i	0.958144	−	0.286285i	$-0.0924205\pi$
$72$	0	0
$73$	63.6965i	0.872554i	0.899812	+	0.436277i	$0.143703\pi$
$73$	63.6965i	0.872554i	−0.899812	+	0.436277i	$0.856297\pi$
$74$	− 88.9506i	− 1.20204i
$75$	0	0
$76$	−19.7910	−0.260407
$77$	−37.3055	−0.484487
$78$	0	0
$79$	82.3035	1.04182	0.520908	−	0.853613i	$-0.325593\pi$
$79$	82.3035	1.04182	0.520908	+	0.853613i	$0.325593\pi$
$80$	0	0
$81$	0	0
$82$	92.9631i	1.13370i
$83$	−99.9021	−1.20364	−0.601820	−	0.798632i	$-0.705558\pi$
$83$	−99.9021	−1.20364	−0.601820	+	0.798632i	$0.705558\pi$
$84$	0	0
$85$	0	0
$86$	71.9014i	0.836063i
$87$	0	0
$88$	− 39.8812i	− 0.453196i
$89$	88.5703i	0.995172i	0.867415	+	0.497586i	$0.165780\pi$
$89$	88.5703i	0.995172i	−0.867415	+	0.497586i	$0.834220\pi$
$90$	0	0
$91$	4.71895	0.0518566
$92$	19.1625	0.208288
$93$	0	0
$94$	92.4054	0.983036
$95$	0	0
$96$	0	0
$97$	117.848i	1.21493i	0.794348	+	0.607464i	$0.207813\pi$
$97$	117.848i	1.21493i	−0.794348	+	0.607464i	$0.792187\pi$
$98$	9.89949	0.101015
$99$	0	0
$100$	0	0
$101$	36.0289i	0.356722i	0.983965	+	0.178361i	$0.0570794\pi$
$101$	36.0289i	0.356722i	−0.983965	+	0.178361i	$0.942921\pi$
$102$	0	0
$103$	− 34.8611i	− 0.338457i	−0.985577	−	0.169229i	$-0.945872\pi$
$103$	− 34.8611i	− 0.338457i	0.985577	−	0.169229i	$-0.0541276\pi$
$104$	5.04477i	0.0485074i
$105$	0	0
$106$	−91.8638	−0.866639
$107$	81.8068	0.764550	0.382275	−	0.924049i	$-0.375141\pi$
$107$	81.8068	0.764550	0.382275	+	0.924049i	$0.375141\pi$
$108$	0	0
$109$	−103.430	−0.948903	−0.474452	−	0.880282i	$-0.657354\pi$
$109$	−103.430	−0.948903	−0.474452	+	0.880282i	$0.657354\pi$
$110$	0	0
$111$	0	0
$112$	10.5830i	0.0944911i
$113$	187.400	1.65841	0.829204	−	0.558946i	$-0.188794\pi$
$113$	187.400	1.65841	0.829204	+	0.558946i	$0.188794\pi$
$114$	0	0
$115$	0	0
$116$	− 81.1763i	− 0.699795i
$117$	0	0
$118$	157.885i	1.33801i
$119$	85.3860i	0.717530i
$120$	0	0
$121$	−77.8140	−0.643091
$122$	−3.80572	−0.0311944
$123$	0	0
$124$	97.1251	0.783267
$125$	0	0
$126$	0	0
$127$	− 81.3040i	− 0.640189i	−0.947386	−	0.320094i	$-0.896285\pi$
$127$	− 81.3040i	− 0.640189i	0.947386	−	0.320094i	$-0.103715\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	0	0
$131$	9.22925i	0.0704523i	0.999379	+	0.0352261i	$0.0112152\pi$
$131$	9.22925i	0.0704523i	−0.999379	+	0.0352261i	$0.988785\pi$
$132$	0	0
$133$	− 26.1810i	− 0.196849i
$134$	− 102.081i	− 0.761801i
$135$	0	0
$136$	−91.2815	−0.671188
$137$	−27.8171	−0.203044	−0.101522	−	0.994833i	$-0.532371\pi$
$137$	−27.8171	−0.203044	−0.101522	+	0.994833i	$0.532371\pi$
$138$	0	0
$139$	101.074	0.727147	0.363574	−	0.931565i	$-0.381556\pi$
$139$	101.074	0.727147	0.363574	+	0.931565i	$0.381556\pi$
$140$	0	0
$141$	0	0
$142$	57.4913i	0.404868i
$143$	25.1489	0.175867
$144$	0	0
$145$	0	0
$146$	− 90.0804i	− 0.616989i
$147$	0	0
$148$	125.795i	0.849967i
$149$	119.738i	0.803612i	0.915725	+	0.401806i	$0.131617\pi$
$149$	119.738i	0.803612i	−0.915725	+	0.401806i	$0.868383\pi$
$150$	0	0
$151$	41.9026	0.277501	0.138750	−	0.990327i	$-0.455691\pi$
$151$	41.9026	0.277501	0.138750	+	0.990327i	$0.455691\pi$
$152$	27.9886	0.184136
$153$	0	0
$154$	52.7579	0.342584
$155$	0	0
$156$	0	0
$157$	130.984i	0.834295i	0.908839	+	0.417148i	$0.136970\pi$
$157$	130.984i	0.834295i	−0.908839	+	0.417148i	$0.863030\pi$
$158$	−116.395	−0.736676
$159$	0	0
$160$	0	0
$161$	25.3496i	0.157451i
$162$	0	0
$163$	− 117.726i	− 0.722246i	−0.932518	−	0.361123i	$-0.882393\pi$
$163$	− 117.726i	− 0.722246i	0.932518	−	0.361123i	$-0.117607\pi$
$164$	− 131.470i	− 0.801645i
$165$	0	0
$166$	141.283	0.851102
$167$	208.939	1.25113	0.625567	−	0.780170i	$-0.284868\pi$
$167$	208.939	1.25113	0.625567	+	0.780170i	$0.284868\pi$
$168$	0	0
$169$	165.819	0.981176
$170$	0	0
$171$	0	0
$172$	− 101.684i	− 0.591186i
$173$	−65.6170	−0.379289	−0.189644	−	0.981853i	$-0.560734\pi$
$173$	−65.6170	−0.379289	−0.189644	+	0.981853i	$0.560734\pi$
$174$	0	0
$175$	0	0
$176$	56.4006i	0.320458i
$177$	0	0
$178$	− 125.257i	− 0.703693i
$179$	− 220.904i	− 1.23410i	−0.786924	−	0.617050i	$-0.788328\pi$
$179$	− 220.904i	− 1.23410i	0.786924	−	0.617050i	$-0.211672\pi$
$180$	0	0
$181$	−84.5556	−0.467158	−0.233579	−	0.972338i	$-0.575044\pi$
$181$	−84.5556	−0.467158	−0.233579	+	0.972338i	$0.575044\pi$
$182$	−6.67360	−0.0366681
$183$	0	0
$184$	−27.0999	−0.147282
$185$	0	0
$186$	0	0
$187$	455.052i	2.43344i
$188$	−130.681	−0.695111
$189$	0	0
$190$	0	0
$191$	− 308.870i	− 1.61712i	−0.588415	−	0.808559i	$-0.700248\pi$
$191$	− 308.870i	− 1.61712i	0.588415	−	0.808559i	$-0.299752\pi$
$192$	0	0
$193$	314.793i	1.63105i	0.578720	+	0.815526i	$0.303552\pi$
$193$	314.793i	1.63105i	−0.578720	+	0.815526i	$0.696448\pi$
$194$	− 166.662i	− 0.859083i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	317.747	1.61293	0.806465	−	0.591282i	$-0.201378\pi$
$197$	317.747	1.61293	0.806465	+	0.591282i	$0.201378\pi$
$198$	0	0
$199$	−123.082	−0.618502	−0.309251	−	0.950980i	$-0.600078\pi$
$199$	−123.082	−0.618502	−0.309251	+	0.950980i	$0.600078\pi$
$200$	0	0
$201$	0	0
$202$	− 50.9526i	− 0.252240i
$203$	107.386	0.528996
$204$	0	0
$205$	0	0
$206$	49.3010i	0.239325i
$207$	0	0
$208$	− 7.13438i	− 0.0342999i
$209$	− 139.528i	− 0.667596i
$210$	0	0
$211$	−298.146	−1.41302	−0.706508	−	0.707705i	$-0.749730\pi$
$211$	−298.146	−1.41302	−0.706508	+	0.707705i	$0.749730\pi$
$212$	129.915	0.612807
$213$	0	0
$214$	−115.692	−0.540618
$215$	0	0
$216$	0	0
$217$	128.484i	0.592094i
$218$	146.273	0.670976
$219$	0	0
$220$	0	0
$221$	− 57.5618i	− 0.260460i
$222$	0	0
$223$	− 83.8687i	− 0.376093i	−0.982160	−	0.188046i	$-0.939784\pi$
$223$	− 83.8687i	− 0.376093i	0.982160	−	0.188046i	$-0.0602155\pi$
$224$	− 14.9666i	− 0.0668153i
$225$	0	0
$226$	−265.024	−1.17267
$227$	−94.6100	−0.416784	−0.208392	−	0.978045i	$-0.566823\pi$
$227$	−94.6100	−0.416784	−0.208392	+	0.978045i	$0.566823\pi$
$228$	0	0
$229$	318.779	1.39205	0.696025	−	0.718018i	$-0.254950\pi$
$229$	318.779	1.39205	0.696025	+	0.718018i	$0.254950\pi$
$230$	0	0
$231$	0	0
$232$	114.801i	0.494830i
$233$	−187.832	−0.806147	−0.403073	−	0.915168i	$-0.632058\pi$
$233$	−187.832	−0.806147	−0.403073	+	0.915168i	$0.632058\pi$
$234$	0	0
$235$	0	0
$236$	− 223.284i	− 0.946117i
$237$	0	0
$238$	− 120.754i	− 0.507370i
$239$	361.090i	1.51084i	0.655243	+	0.755418i	$0.272566\pi$
$239$	361.090i	1.51084i	−0.655243	+	0.755418i	$0.727434\pi$
$240$	0	0
$241$	−14.5899	−0.0605391	−0.0302695	−	0.999542i	$-0.509637\pi$
$241$	−14.5899	−0.0605391	−0.0302695	+	0.999542i	$0.509637\pi$
$242$	110.046	0.454734
$243$	0	0
$244$	5.38210	0.0220578
$245$	0	0
$246$	0	0
$247$	17.6495i	0.0714556i
$248$	−137.356	−0.553854
$249$	0	0
$250$	0	0
$251$	− 359.495i	− 1.43225i	−0.697971	−	0.716126i	$-0.745914\pi$
$251$	− 359.495i	− 1.43225i	0.697971	−	0.716126i	$-0.254086\pi$
$252$	0	0
$253$	135.097i	0.533981i
$254$	114.981i	0.452682i
$255$	0	0
$256$	16.0000	0.0625000
$257$	−196.720	−0.765447	−0.382724	−	0.923863i	$-0.625014\pi$
$257$	−196.720	−0.765447	−0.382724	+	0.923863i	$0.625014\pi$
$258$	0	0
$259$	−166.411	−0.642515
$260$	0	0
$261$	0	0
$262$	− 13.0521i	− 0.0498173i
$263$	396.028	1.50581	0.752905	−	0.658129i	$-0.228652\pi$
$263$	396.028	1.50581	0.752905	+	0.658129i	$0.228652\pi$
$264$	0	0
$265$	0	0
$266$	37.0255i	0.139194i
$267$	0	0
$268$	144.365i	0.538674i
$269$	− 203.497i	− 0.756494i	−0.925705	−	0.378247i	$-0.876527\pi$
$269$	− 203.497i	− 0.756494i	0.925705	−	0.378247i	$-0.123473\pi$
$270$	0	0
$271$	−173.701	−0.640964	−0.320482	−	0.947255i	$-0.603845\pi$
$271$	−173.701	−0.640964	−0.320482	+	0.947255i	$0.603845\pi$
$272$	129.092	0.474601
$273$	0	0
$274$	39.3393	0.143574
$275$	0	0
$276$	0	0
$277$	− 130.018i	− 0.469380i	−0.972070	−	0.234690i	$-0.924593\pi$
$277$	− 130.018i	− 0.469380i	0.972070	−	0.234690i	$-0.0754075\pi$
$278$	−142.940	−0.514171
$279$	0	0
$280$	0	0
$281$	476.464i	1.69560i	0.530314	+	0.847801i	$0.322074\pi$
$281$	476.464i	1.69560i	−0.530314	+	0.847801i	$0.677926\pi$
$282$	0	0
$283$	56.5824i	0.199938i	0.994991	+	0.0999690i	$0.0318744\pi$
$283$	56.5824i	0.199938i	−0.994991	+	0.0999690i	$0.968126\pi$
$284$	− 81.3050i	− 0.286285i
$285$	0	0
$286$	−35.5660	−0.124357
$287$	173.918	0.605986
$288$	0	0
$289$	752.540	2.60394
$290$	0	0
$291$	0	0
$292$	127.393i	0.436277i
$293$	−124.651	−0.425431	−0.212715	−	0.977114i	$-0.568231\pi$
$293$	−124.651	−0.425431	−0.212715	+	0.977114i	$0.568231\pi$
$294$	0	0
$295$	0	0
$296$	− 177.901i	− 0.601018i
$297$	0	0
$298$	− 169.335i	− 0.568239i
$299$	− 17.0891i	− 0.0571542i
$300$	0	0
$301$	134.515	0.446895
$302$	−59.2592	−0.196223
$303$	0	0
$304$	−39.5819	−0.130204
$305$	0	0
$306$	0	0
$307$	27.2322i	0.0887042i	0.999016	+	0.0443521i	$0.0141223\pi$
$307$	27.2322i	0.0887042i	−0.999016	+	0.0443521i	$0.985878\pi$
$308$	−74.6109	−0.242243
$309$	0	0
$310$	0	0
$311$	63.9207i	0.205533i	0.994706	+	0.102766i	$0.0327694\pi$
$311$	63.9207i	0.205533i	−0.994706	+	0.102766i	$0.967231\pi$
$312$	0	0
$313$	− 277.105i	− 0.885320i	−0.896689	−	0.442660i	$-0.854035\pi$
$313$	− 277.105i	− 0.885320i	0.896689	−	0.442660i	$-0.145965\pi$
$314$	− 185.240i	− 0.589936i
$315$	0	0
$316$	164.607	0.520908
$317$	407.861	1.28663	0.643313	−	0.765603i	$-0.277559\pi$
$317$	407.861	1.28663	0.643313	+	0.765603i	$0.277559\pi$
$318$	0	0
$319$	572.299	1.79404
$320$	0	0
$321$	0	0
$322$	− 35.8498i	− 0.111335i
$323$	−319.356	−0.988717
$324$	0	0
$325$	0	0
$326$	166.490i	0.510705i
$327$	0	0
$328$	185.926i	0.566848i
$329$	− 172.875i	− 0.525455i
$330$	0	0
$331$	350.438	1.05872	0.529362	−	0.848396i	$-0.322431\pi$
$331$	350.438	1.05872	0.529362	+	0.848396i	$0.322431\pi$
$332$	−199.804	−0.601820
$333$	0	0
$334$	−295.485	−0.884686
$335$	0	0
$336$	0	0
$337$	506.953i	1.50431i	0.658985	+	0.752156i	$0.270986\pi$
$337$	506.953i	1.50431i	−0.658985	+	0.752156i	$0.729014\pi$
$338$	−234.503	−0.693796
$339$	0	0
$340$	0	0
$341$	684.739i	2.00803i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	143.803i	0.418032i
$345$	0	0
$346$	92.7964	0.268198
$347$	143.703	0.414131	0.207065	−	0.978327i	$-0.433609\pi$
$347$	143.703	0.414131	0.207065	+	0.978327i	$0.433609\pi$
$348$	0	0
$349$	168.383	0.482473	0.241237	−	0.970466i	$-0.422447\pi$
$349$	168.383	0.482473	0.241237	+	0.970466i	$0.422447\pi$
$350$	0	0
$351$	0	0
$352$	− 79.7625i	− 0.226598i
$353$	531.562	1.50584	0.752921	−	0.658111i	$-0.228644\pi$
$353$	531.562	1.50584	0.752921	+	0.658111i	$0.228644\pi$
$354$	0	0
$355$	0	0
$356$	177.141i	0.497586i
$357$	0	0
$358$	312.405i	0.872640i
$359$	63.2884i	0.176291i	0.996108	+	0.0881454i	$0.0280940\pi$
$359$	63.2884i	0.176291i	−0.996108	+	0.0881454i	$0.971906\pi$
$360$	0	0
$361$	−263.080	−0.728752
$362$	119.580	0.330331
$363$	0	0
$364$	9.43790	0.0259283
$365$	0	0
$366$	0	0
$367$	252.555i	0.688160i	0.938940	+	0.344080i	$0.111809\pi$
$367$	252.555i	0.688160i	−0.938940	+	0.344080i	$0.888191\pi$
$368$	38.3251	0.104144
$369$	0	0
$370$	0	0
$371$	171.861i	0.463238i
$372$	0	0
$373$	223.925i	0.600336i	0.953886	+	0.300168i	$0.0970427\pi$
$373$	223.925i	0.600336i	−0.953886	+	0.300168i	$0.902957\pi$
$374$	− 643.541i	− 1.72070i
$375$	0	0
$376$	184.811	0.491518
$377$	−72.3928	−0.192023
$378$	0	0
$379$	503.313	1.32800	0.664001	−	0.747732i	$-0.268857\pi$
$379$	503.313	1.32800	0.664001	+	0.747732i	$0.268857\pi$
$380$	0	0
$381$	0	0
$382$	436.808i	1.14348i
$383$	446.558	1.16595	0.582974	−	0.812491i	$-0.301889\pi$
$383$	446.558	1.16595	0.582974	+	0.812491i	$0.301889\pi$
$384$	0	0
$385$	0	0
$386$	− 445.185i	− 1.15333i
$387$	0	0
$388$	235.696i	0.607464i
$389$	− 18.4450i	− 0.0474165i	−0.999719	−	0.0237083i	$-0.992453\pi$
$389$	− 18.4450i	− 0.0474165i	0.999719	−	0.0237083i	$-0.00754728\pi$
$390$	0	0
$391$	309.215	0.790831
$392$	19.7990	0.0505076
$393$	0	0
$394$	−449.362	−1.14051
$395$	0	0
$396$	0	0
$397$	− 209.590i	− 0.527934i	−0.964532	−	0.263967i	$-0.914969\pi$
$397$	− 209.590i	− 0.527934i	0.964532	−	0.263967i	$-0.0850310\pi$
$398$	174.064	0.437347
$399$	0	0
$400$	0	0
$401$	277.998i	0.693261i	0.938002	+	0.346631i	$0.112674\pi$
$401$	277.998i	0.693261i	−0.938002	+	0.346631i	$0.887326\pi$
$402$	0	0
$403$	− 86.6160i	− 0.214928i
$404$	72.0578i	0.178361i
$405$	0	0
$406$	−151.867	−0.374056
$407$	−886.865	−2.17903
$408$	0	0
$409$	506.079	1.23736	0.618678	−	0.785644i	$-0.287668\pi$
$409$	506.079	1.23736	0.618678	+	0.785644i	$0.287668\pi$
$410$	0	0
$411$	0	0
$412$	− 69.7221i	− 0.169229i
$413$	295.377	0.715198
$414$	0	0
$415$	0	0
$416$	10.0895i	0.0242537i
$417$	0	0
$418$	197.322i	0.472062i
$419$	95.9760i	0.229060i	0.993420	+	0.114530i	$0.0365362\pi$
$419$	95.9760i	0.229060i	−0.993420	+	0.114530i	$0.963464\pi$
$420$	0	0
$421$	−236.741	−0.562330	−0.281165	−	0.959659i	$-0.590721\pi$
$421$	−236.741	−0.562330	−0.281165	+	0.959659i	$0.590721\pi$
$422$	421.642	0.999153
$423$	0	0
$424$	−183.728	−0.433320
$425$	0	0
$426$	0	0
$427$	7.11985i	0.0166741i
$428$	163.614	0.382275
$429$	0	0
$430$	0	0
$431$	494.406i	1.14711i	0.819166	+	0.573557i	$0.194437\pi$
$431$	494.406i	1.14711i	−0.819166	+	0.573557i	$0.805563\pi$
$432$	0	0
$433$	426.179i	0.984247i	0.870526	+	0.492123i	$0.163779\pi$
$433$	426.179i	0.984247i	−0.870526	+	0.492123i	$0.836221\pi$
$434$	− 181.705i	− 0.418674i
$435$	0	0
$436$	−206.861	−0.474452
$437$	−94.8112	−0.216959
$438$	0	0
$439$	−374.522	−0.853125	−0.426562	−	0.904458i	$-0.640276\pi$
$439$	−374.522	−0.853125	−0.426562	+	0.904458i	$0.640276\pi$
$440$	0	0
$441$	0	0
$442$	81.4046i	0.184173i
$443$	−38.1307	−0.0860739	−0.0430370	−	0.999073i	$-0.513703\pi$
$443$	−38.1307	−0.0860739	−0.0430370	+	0.999073i	$0.513703\pi$
$444$	0	0
$445$	0	0
$446$	118.608i	0.265938i
$447$	0	0
$448$	21.1660i	0.0472456i
$449$	− 1.52774i	− 0.00340253i	−0.999999	−	0.00170126i	$-0.999458\pi$
$449$	− 1.52774i	− 0.00340253i	0.999999	−	0.00170126i	$-0.000541529\pi$
$450$	0	0
$451$	926.871	2.05515
$452$	374.800	0.829204
$453$	0	0
$454$	133.799	0.294711
$455$	0	0
$456$	0	0
$457$	− 141.016i	− 0.308569i	−0.988026	−	0.154284i	$-0.950693\pi$
$457$	− 141.016i	− 0.308569i	0.988026	−	0.154284i	$-0.0493072\pi$
$458$	−450.822	−0.984327
$459$	0	0
$460$	0	0
$461$	− 10.4463i	− 0.0226601i	−0.999936	−	0.0113301i	$-0.996393\pi$
$461$	− 10.4463i	− 0.0226601i	0.999936	−	0.0113301i	$-0.00360655\pi$
$462$	0	0
$463$	− 159.805i	− 0.345152i	−0.984996	−	0.172576i	$-0.944791\pi$
$463$	− 159.805i	− 0.345152i	0.984996	−	0.172576i	$-0.0552091\pi$
$464$	− 162.353i	− 0.349898i
$465$	0	0
$466$	265.635	0.570032
$467$	−58.0764	−0.124361	−0.0621803	−	0.998065i	$-0.519805\pi$
$467$	−58.0764	−0.124361	−0.0621803	+	0.998065i	$0.519805\pi$
$468$	0	0
$469$	−190.977	−0.407200
$470$	0	0
$471$	0	0
$472$	315.771i	0.669006i
$473$	716.879	1.51560
$474$	0	0
$475$	0	0
$476$	170.772i	0.358765i
$477$	0	0
$478$	− 510.658i	− 1.06832i
$479$	134.072i	0.279899i	0.990159	+	0.139950i	$0.0446940\pi$
$479$	134.072i	0.279899i	−0.990159	+	0.139950i	$0.955306\pi$
$480$	0	0
$481$	112.184	0.233230
$482$	20.6333	0.0428076
$483$	0	0
$484$	−155.628	−0.321546
$485$	0	0
$486$	0	0
$487$	277.859i	0.570552i	0.958445	+	0.285276i	$0.0920853\pi$
$487$	277.859i	0.570552i	−0.958445	+	0.285276i	$0.907915\pi$
$488$	−7.61144	−0.0155972
$489$	0	0
$490$	0	0
$491$	− 514.975i	− 1.04883i	−0.851463	−	0.524414i	$-0.824284\pi$
$491$	− 514.975i	− 1.04883i	0.851463	−	0.524414i	$-0.175716\pi$
$492$	0	0
$493$	− 1309.90i	− 2.65699i
$494$	− 24.9602i	− 0.0505267i
$495$	0	0
$496$	194.250	0.391634
$497$	107.556	0.216411
$498$	0	0
$499$	144.052	0.288682	0.144341	−	0.989528i	$-0.453894\pi$
$499$	144.052	0.288682	0.144341	+	0.989528i	$0.453894\pi$
$500$	0	0
$501$	0	0
$502$	508.403i	1.01275i
$503$	596.609	1.18610	0.593051	−	0.805165i	$-0.297923\pi$
$503$	596.609	1.18610	0.593051	+	0.805165i	$0.297923\pi$
$504$	0	0
$505$	0	0
$506$	− 191.056i	− 0.377582i
$507$	0	0
$508$	− 162.608i	− 0.320094i
$509$	− 7.91201i	− 0.0155442i	−0.999970	−	0.00777211i	$-0.997526\pi$
$509$	− 7.91201i	− 0.0155442i	0.999970	−	0.00777211i	$-0.00247397\pi$
$510$	0	0
$511$	−168.525	−0.329794
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	278.204	0.541253
$515$	0	0
$516$	0	0
$517$	− 921.310i	− 1.78203i
$518$	235.341	0.454327
$519$	0	0
$520$	0	0
$521$	746.931i	1.43365i	0.697253	+	0.716825i	$0.254405\pi$
$521$	746.931i	1.43365i	−0.697253	+	0.716825i	$0.745595\pi$
$522$	0	0
$523$	138.896i	0.265576i	0.991144	+	0.132788i	$0.0423929\pi$
$523$	138.896i	0.265576i	−0.991144	+	0.132788i	$0.957607\pi$
$524$	18.4585i	0.0352261i
$525$	0	0
$526$	−560.068	−1.06477
$527$	1567.25	2.97392
$528$	0	0
$529$	−437.199	−0.826464
$530$	0	0
$531$	0	0
$532$	− 52.3619i	− 0.0984247i
$533$	−117.244	−0.219971
$534$	0	0
$535$	0	0
$536$	− 204.163i	− 0.380900i
$537$	0	0
$538$	287.788i	0.534922i
$539$	− 98.7010i	− 0.183119i
$540$	0	0
$541$	−684.033	−1.26439	−0.632193	−	0.774811i	$-0.717845\pi$
$541$	−684.033	−1.26439	−0.632193	+	0.774811i	$0.717845\pi$
$542$	245.651	0.453230
$543$	0	0
$544$	−182.563	−0.335594
$545$	0	0
$546$	0	0
$547$	485.135i	0.886901i	0.896299	+	0.443451i	$0.146246\pi$
$547$	485.135i	0.886901i	−0.896299	+	0.443451i	$0.853754\pi$
$548$	−55.6341	−0.101522
$549$	0	0
$550$	0	0
$551$	401.639i	0.728927i
$552$	0	0
$553$	217.755i	0.393770i
$554$	183.874i	0.331902i
$555$	0	0
$556$	202.147	0.363574
$557$	−227.641	−0.408692	−0.204346	−	0.978899i	$-0.565507\pi$
$557$	−227.641	−0.408692	−0.204346	+	0.978899i	$0.565507\pi$
$558$	0	0
$559$	−90.6815	−0.162221
$560$	0	0
$561$	0	0
$562$	− 673.822i	− 1.19897i
$563$	−69.7030	−0.123806	−0.0619032	−	0.998082i	$-0.519717\pi$
$563$	−69.7030	−0.123806	−0.0619032	+	0.998082i	$0.519717\pi$
$564$	0	0
$565$	0	0
$566$	− 80.0197i	− 0.141378i
$567$	0	0
$568$	114.983i	0.202434i
$569$	− 573.975i	− 1.00874i	−0.863487	−	0.504372i	$-0.831724\pi$
$569$	− 573.975i	− 1.00874i	0.863487	−	0.504372i	$-0.168276\pi$
$570$	0	0
$571$	289.675	0.507312	0.253656	−	0.967294i	$-0.418367\pi$
$571$	289.675	0.507312	0.253656	+	0.967294i	$0.418367\pi$
$572$	50.2979	0.0879334
$573$	0	0
$574$	−245.957	−0.428497
$575$	0	0
$576$	0	0
$577$	122.652i	0.212569i	0.994336	+	0.106285i	$0.0338955\pi$
$577$	122.652i	0.212569i	−0.994336	+	0.106285i	$0.966105\pi$
$578$	−1064.25	−1.84127
$579$	0	0
$580$	0	0
$581$	− 264.316i	− 0.454933i
$582$	0	0
$583$	915.910i	1.57103i
$584$	− 180.161i	− 0.308494i
$585$	0	0
$586$	176.283	0.300825
$587$	156.694	0.266940	0.133470	−	0.991053i	$-0.457388\pi$
$587$	156.694	0.266940	0.133470	+	0.991053i	$0.457388\pi$
$588$	0	0
$589$	−480.550	−0.815874
$590$	0	0
$591$	0	0
$592$	251.590i	0.424984i
$593$	−78.4823	−0.132348	−0.0661739	−	0.997808i	$-0.521079\pi$
$593$	−78.4823	−0.132348	−0.0661739	+	0.997808i	$0.521079\pi$
$594$	0	0
$595$	0	0
$596$	239.476i	0.401806i
$597$	0	0
$598$	24.1676i	0.0404141i
$599$	104.010i	0.173639i	0.996224	+	0.0868195i	$0.0276703\pi$
$599$	104.010i	0.173639i	−0.996224	+	0.0868195i	$0.972330\pi$
$600$	0	0
$601$	773.253	1.28661	0.643305	−	0.765610i	$-0.277563\pi$
$601$	773.253	1.28661	0.643305	+	0.765610i	$0.277563\pi$
$602$	−190.233	−0.316002
$603$	0	0
$604$	83.8052	0.138750
$605$	0	0
$606$	0	0
$607$	349.051i	0.575044i	0.957774	+	0.287522i	$0.0928314\pi$
$607$	349.051i	0.575044i	−0.957774	+	0.287522i	$0.907169\pi$
$608$	55.9773	0.0920679
$609$	0	0
$610$	0	0
$611$	116.541i	0.190738i
$612$	0	0
$613$	323.779i	0.528188i	0.964497	+	0.264094i	$0.0850729\pi$
$613$	323.779i	0.528188i	−0.964497	+	0.264094i	$0.914927\pi$
$614$	− 38.5121i	− 0.0627233i
$615$	0	0
$616$	105.516	0.171292
$617$	1040.96	1.68713	0.843567	−	0.537023i	$-0.180451\pi$
$617$	1040.96	1.68713	0.843567	+	0.537023i	$0.180451\pi$
$618$	0	0
$619$	783.600	1.26591	0.632957	−	0.774187i	$-0.281841\pi$
$619$	783.600	1.26591	0.632957	+	0.774187i	$0.281841\pi$
$620$	0	0
$621$	0	0
$622$	− 90.3976i	− 0.145334i
$623$	−234.335	−0.376140
$624$	0	0
$625$	0	0
$626$	391.886i	0.626016i
$627$	0	0
$628$	261.969i	0.417148i
$629$	2029.89i	3.22717i
$630$	0	0
$631$	967.921	1.53395	0.766974	−	0.641678i	$-0.221762\pi$
$631$	967.921	1.53395	0.766974	+	0.641678i	$0.221762\pi$
$632$	−232.790	−0.368338
$633$	0	0
$634$	−576.802	−0.909783
$635$	0	0
$636$	0	0
$637$	12.4852i	0.0195999i
$638$	−809.352	−1.26858
$639$	0	0
$640$	0	0
$641$	− 38.9129i	− 0.0607066i	−0.999539	−	0.0303533i	$-0.990337\pi$
$641$	− 38.9129i	− 0.0607066i	0.999539	−	0.0303533i	$-0.00966324\pi$
$642$	0	0
$643$	− 1123.77i	− 1.74771i	−0.486191	−	0.873853i	$-0.661614\pi$
$643$	− 1123.77i	− 1.74771i	0.486191	−	0.873853i	$-0.338386\pi$
$644$	50.6993i	0.0787256i
$645$	0	0
$646$	451.637	0.699129
$647$	635.461	0.982165	0.491083	−	0.871113i	$-0.336601\pi$
$647$	635.461	0.982165	0.491083	+	0.871113i	$0.336601\pi$
$648$	0	0
$649$	1574.17	2.42553
$650$	0	0
$651$	0	0
$652$	− 235.452i	− 0.361123i
$653$	−361.063	−0.552929	−0.276465	−	0.961024i	$-0.589163\pi$
$653$	−361.063	−0.552929	−0.276465	+	0.961024i	$0.589163\pi$
$654$	0	0
$655$	0	0
$656$	− 262.939i	− 0.400822i
$657$	0	0
$658$	244.482i	0.371553i
$659$	1284.71i	1.94949i	0.223327	+	0.974744i	$0.428308\pi$
$659$	1284.71i	1.94949i	−0.223327	+	0.974744i	$0.571692\pi$
$660$	0	0
$661$	605.603	0.916192	0.458096	−	0.888903i	$-0.348532\pi$
$661$	605.603	0.916192	0.458096	+	0.888903i	$0.348532\pi$
$662$	−495.594	−0.748632
$663$	0	0
$664$	282.566	0.425551
$665$	0	0
$666$	0	0
$667$	− 388.886i	− 0.583037i
$668$	417.879	0.625567
$669$	0	0
$670$	0	0
$671$	37.9442i	0.0565487i
$672$	0	0
$673$	863.984i	1.28378i	0.766797	+	0.641890i	$0.221849\pi$
$673$	863.984i	1.28378i	−0.766797	+	0.641890i	$0.778151\pi$
$674$	− 716.940i	− 1.06371i
$675$	0	0
$676$	331.638	0.490588
$677$	−1009.59	−1.49127	−0.745635	−	0.666354i	$-0.767854\pi$
$677$	−1009.59	−1.49127	−0.745635	+	0.666354i	$0.767854\pi$
$678$	0	0
$679$	−311.796	−0.459199
$680$	0	0
$681$	0	0
$682$	− 968.367i	− 1.41989i
$683$	786.887	1.15210	0.576052	−	0.817413i	$-0.304593\pi$
$683$	786.887	1.15210	0.576052	+	0.817413i	$0.304593\pi$
$684$	0	0
$685$	0	0
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	− 203.368i	− 0.295593i
$689$	− 115.858i	− 0.168154i
$690$	0	0
$691$	−1285.82	−1.86081	−0.930405	−	0.366534i	$-0.880544\pi$
$691$	−1285.82	−1.86081	−0.930405	+	0.366534i	$0.880544\pi$
$692$	−131.234	−0.189644
$693$	0	0
$694$	−203.227	−0.292835
$695$	0	0
$696$	0	0
$697$	− 2121.45i	− 3.04369i
$698$	−238.130	−0.341160
$699$	0	0
$700$	0	0
$701$	717.153i	1.02304i	0.859271	+	0.511521i	$0.170918\pi$
$701$	717.153i	1.02304i	−0.859271	+	0.511521i	$0.829082\pi$
$702$	0	0
$703$	− 622.402i	− 0.885351i
$704$	112.801i	0.160229i
$705$	0	0
$706$	−751.742	−1.06479
$707$	−95.3235	−0.134828
$708$	0	0
$709$	−197.923	−0.279158	−0.139579	−	0.990211i	$-0.544575\pi$
$709$	−197.923	−0.279158	−0.139579	+	0.990211i	$0.544575\pi$
$710$	0	0
$711$	0	0
$712$	− 250.515i	− 0.351847i
$713$	465.291	0.652582
$714$	0	0
$715$	0	0
$716$	− 441.808i	− 0.617050i
$717$	0	0
$718$	− 89.5033i	− 0.124656i
$719$	− 168.511i	− 0.234369i	−0.993110	−	0.117185i	$-0.962613\pi$
$719$	− 168.511i	− 0.234369i	0.993110	−	0.117185i	$-0.0373869\pi$
$720$	0	0
$721$	92.2337	0.127925
$722$	372.051	0.515306
$723$	0	0
$724$	−169.111	−0.233579
$725$	0	0
$726$	0	0
$727$	906.873i	1.24742i	0.781657	+	0.623709i	$0.214375\pi$
$727$	906.873i	1.24742i	−0.781657	+	0.623709i	$0.785625\pi$
$728$	−13.3472	−0.0183341
$729$	0	0
$730$	0	0
$731$	− 1640.82i	− 2.24462i
$732$	0	0
$733$	22.6632i	0.0309184i	0.999880	+	0.0154592i	$0.00492101\pi$
$733$	22.6632i	0.0309184i	−0.999880	+	0.0154592i	$0.995079\pi$
$734$	− 357.166i	− 0.486602i
$735$	0	0
$736$	−54.1998	−0.0736410
$737$	−1017.78	−1.38098
$738$	0	0
$739$	−614.258	−0.831202	−0.415601	−	0.909547i	$-0.636429\pi$
$739$	−614.258	−0.831202	−0.415601	+	0.909547i	$0.636429\pi$
$740$	0	0
$741$	0	0
$742$	− 243.049i	− 0.327559i
$743$	33.0291	0.0444537	0.0222268	−	0.999753i	$-0.492924\pi$
$743$	33.0291	0.0444537	0.0222268	+	0.999753i	$0.492924\pi$
$744$	0	0
$745$	0	0
$746$	− 316.678i	− 0.424501i
$747$	0	0
$748$	910.105i	1.21672i
$749$	216.441i	0.288973i
$750$	0	0
$751$	−905.403	−1.20560	−0.602798	−	0.797894i	$-0.705948\pi$
$751$	−905.403	−1.20560	−0.602798	+	0.797894i	$0.705948\pi$
$752$	−261.362	−0.347556
$753$	0	0
$754$	102.379	0.135781
$755$	0	0
$756$	0	0
$757$	64.4031i	0.0850768i	0.999095	+	0.0425384i	$0.0135445\pi$
$757$	64.4031i	0.0850768i	−0.999095	+	0.0425384i	$0.986456\pi$
$758$	−711.791	−0.939039
$759$	0	0
$760$	0	0
$761$	− 508.718i	− 0.668486i	−0.942487	−	0.334243i	$-0.891519\pi$
$761$	− 508.718i	− 0.668486i	0.942487	−	0.334243i	$-0.108481\pi$
$762$	0	0
$763$	− 273.651i	− 0.358652i
$764$	− 617.739i	− 0.808559i
$765$	0	0
$766$	−631.528	−0.824449
$767$	−199.124	−0.259614
$768$	0	0
$769$	934.507	1.21522	0.607612	−	0.794234i	$-0.292128\pi$
$769$	934.507	1.21522	0.607612	+	0.794234i	$0.292128\pi$
$770$	0	0
$771$	0	0
$772$	629.586i	0.815526i
$773$	−1013.90	−1.31164	−0.655821	−	0.754917i	$-0.727677\pi$
$773$	−1013.90	−1.31164	−0.655821	+	0.754917i	$0.727677\pi$
$774$	0	0
$775$	0	0
$776$	− 333.324i	− 0.429542i
$777$	0	0
$778$	26.0852i	0.0335285i
$779$	650.478i	0.835017i
$780$	0	0
$781$	573.206	0.733939
$782$	−437.296	−0.559202
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	374.605i	0.475991i	0.971266	+	0.237996i	$0.0764904\pi$
$787$	374.605i	0.475991i	−0.971266	+	0.237996i	$0.923510\pi$
$788$	635.494	0.806465
$789$	0	0
$790$	0	0
$791$	495.814i	0.626819i
$792$	0	0
$793$	− 4.79975i	− 0.00605264i
$794$	296.405i	0.373306i
$795$	0	0
$796$	−246.164	−0.309251
$797$	−734.515	−0.921600	−0.460800	−	0.887504i	$-0.652437\pi$
$797$	−734.515	−0.921600	−0.460800	+	0.887504i	$0.652437\pi$
$798$	0	0
$799$	−2108.73	−2.63921
$800$	0	0
$801$	0	0
$802$	− 393.148i	− 0.490210i
$803$	−898.129	−1.11847
$804$	0	0
$805$	0	0
$806$	122.493i	0.151977i
$807$	0	0
$808$	− 101.905i	− 0.126120i
$809$	212.258i	0.262371i	0.991358	+	0.131186i	$0.0418784\pi$
$809$	212.258i	0.262371i	−0.991358	+	0.131186i	$0.958122\pi$
$810$	0	0
$811$	−1343.78	−1.65694	−0.828470	−	0.560034i	$-0.810788\pi$
$811$	−1343.78	−1.65694	−0.828470	+	0.560034i	$0.810788\pi$
$812$	214.772	0.264498
$813$	0	0
$814$	1254.22	1.54081
$815$	0	0
$816$	0	0
$817$	503.106i	0.615797i
$818$	−715.703	−0.874943
$819$	0	0
$820$	0	0
$821$	126.090i	0.153582i	0.997047	+	0.0767908i	$0.0244674\pi$
$821$	126.090i	0.153582i	−0.997047	+	0.0767908i	$0.975533\pi$
$822$	0	0
$823$	74.0891i	0.0900232i	0.998986	+	0.0450116i	$0.0143325\pi$
$823$	74.0891i	0.0900232i	−0.998986	+	0.0450116i	$0.985668\pi$
$824$	98.6020i	0.119663i
$825$	0	0
$826$	−417.726	−0.505721
$827$	−626.689	−0.757786	−0.378893	−	0.925441i	$-0.623695\pi$
$827$	−626.689	−0.757786	−0.378893	+	0.925441i	$0.623695\pi$
$828$	0	0
$829$	−605.458	−0.730348	−0.365174	−	0.930939i	$-0.618990\pi$
$829$	−605.458	−0.730348	−0.365174	+	0.930939i	$0.618990\pi$
$830$	0	0
$831$	0	0
$832$	− 14.2688i	− 0.0171500i
$833$	−225.910	−0.271201
$834$	0	0
$835$	0	0
$836$	− 279.055i	− 0.333798i
$837$	0	0
$838$	− 135.731i	− 0.161970i
$839$	1303.81i	1.55400i	0.629499	+	0.777001i	$0.283260\pi$
$839$	1303.81i	1.55400i	−0.629499	+	0.777001i	$0.716740\pi$
$840$	0	0
$841$	−806.397	−0.958855
$842$	334.802	0.397627
$843$	0	0
$844$	−596.293	−0.706508
$845$	0	0
$846$	0	0
$847$	− 205.877i	− 0.243066i
$848$	259.830	0.306403
$849$	0	0
$850$	0	0
$851$	602.638i	0.708153i
$852$	0	0
$853$	34.1253i	0.0400062i	0.999800	+	0.0200031i	$0.00636761\pi$
$853$	34.1253i	0.0400062i	−0.999800	+	0.0200031i	$0.993632\pi$
$854$	− 10.0690i	− 0.0117904i
$855$	0	0
$856$	−231.385	−0.270309
$857$	1513.23	1.76573	0.882866	−	0.469626i	$-0.155611\pi$
$857$	1513.23	1.76573	0.882866	+	0.469626i	$0.155611\pi$
$858$	0	0
$859$	1089.68	1.26855	0.634273	−	0.773109i	$-0.281299\pi$
$859$	1089.68	1.26855	0.634273	+	0.773109i	$0.281299\pi$
$860$	0	0
$861$	0	0
$862$	− 699.196i	− 0.811132i
$863$	−643.590	−0.745759	−0.372880	−	0.927880i	$-0.621630\pi$
$863$	−643.590	−0.745759	−0.372880	+	0.927880i	$0.621630\pi$
$864$	0	0
$865$	0	0
$866$	− 602.708i	− 0.695967i
$867$	0	0
$868$	256.969i	0.296047i
$869$	1160.49i	1.33543i
$870$	0	0
$871$	128.744	0.147812
$872$	292.545	0.335488
$873$	0	0
$874$	134.083	0.153413
$875$	0	0
$876$	0	0
$877$	− 766.776i	− 0.874317i	−0.899385	−	0.437158i	$-0.855985\pi$
$877$	− 766.776i	− 0.874317i	0.899385	−	0.437158i	$-0.144015\pi$
$878$	529.654	0.603250
$879$	0	0
$880$	0	0
$881$	− 157.211i	− 0.178446i	−0.996012	−	0.0892229i	$-0.971562\pi$
$881$	− 157.211i	− 0.178446i	0.996012	−	0.0892229i	$-0.0284384\pi$
$882$	0	0
$883$	14.1471i	0.0160216i	0.999968	+	0.00801079i	$0.00254994\pi$
$883$	14.1471i	0.0160216i	−0.999968	+	0.00801079i	$0.997450\pi$
$884$	− 115.124i	− 0.130230i
$885$	0	0
$886$	53.9250	0.0608635
$887$	−1741.26	−1.96309	−0.981547	−	0.191220i	$-0.938756\pi$
$887$	−1741.26	−1.96309	−0.981547	+	0.191220i	$0.938756\pi$
$888$	0	0
$889$	215.110	0.241969
$890$	0	0
$891$	0	0
$892$	− 167.737i	− 0.188046i
$893$	646.575	0.724048
$894$	0	0
$895$	0	0
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	2.16054i	0.00240595i
$899$	− 1971.06i	− 2.19251i
$900$	0	0
$901$	2096.37	2.32671
$902$	−1310.79	−1.45321
$903$	0	0
$904$	−530.047	−0.586336
$905$	0	0
$906$	0	0
$907$	− 1290.67i	− 1.42301i	−0.702681	−	0.711505i	$-0.748014\pi$
$907$	− 1290.67i	− 1.42301i	0.702681	−	0.711505i	$-0.251986\pi$
$908$	−189.220	−0.208392
$909$	0	0
$910$	0	0
$911$	− 906.341i	− 0.994886i	−0.867497	−	0.497443i	$-0.834272\pi$
$911$	− 906.341i	− 0.994886i	0.867497	−	0.497443i	$-0.165728\pi$
$912$	0	0
$913$	− 1408.63i	− 1.54286i
$914$	199.427i	0.218191i
$915$	0	0
$916$	637.559	0.696025
$917$	−24.4183	−0.0266285
$918$	0	0
$919$	−1738.74	−1.89199	−0.945994	−	0.324185i	$-0.894910\pi$
$919$	−1738.74	−1.89199	−0.945994	+	0.324185i	$0.894910\pi$
$920$	0	0
$921$	0	0
$922$	14.7733i	0.0160231i
$923$	−72.5076	−0.0785564
$924$	0	0
$925$	0	0
$926$	225.999i	0.244059i
$927$	0	0
$928$	229.601i	0.247415i
$929$	529.634i	0.570112i	0.958511	+	0.285056i	$0.0920122\pi$
$929$	529.634i	0.570112i	−0.958511	+	0.285056i	$0.907988\pi$
$930$	0	0
$931$	69.2683	0.0744021
$932$	−375.664	−0.403073
$933$	0	0
$934$	82.1324	0.0879362
$935$	0	0
$936$	0	0
$937$	− 1354.47i	− 1.44553i	−0.691091	−	0.722767i	$-0.742870\pi$
$937$	− 1354.47i	− 1.44553i	0.691091	−	0.722767i	$-0.257130\pi$
$938$	270.082	0.287934
$939$	0	0
$940$	0	0
$941$	− 1406.13i	− 1.49429i	−0.664662	−	0.747144i	$-0.731425\pi$
$941$	− 1406.13i	− 1.49429i	0.664662	−	0.747144i	$-0.268575\pi$
$942$	0	0
$943$	− 629.823i	− 0.667893i
$944$	− 446.567i	− 0.473059i
$945$	0	0
$946$	−1013.82	−1.07169
$947$	−1168.66	−1.23406	−0.617032	−	0.786938i	$-0.711665\pi$
$947$	−1168.66	−1.23406	−0.617032	+	0.786938i	$0.711665\pi$
$948$	0	0
$949$	113.609	0.119714
$950$	0	0
$951$	0	0
$952$	− 241.508i	− 0.253685i
$953$	−65.5411	−0.0687735	−0.0343867	−	0.999409i	$-0.510948\pi$
$953$	−65.5411	−0.0687735	−0.0343867	+	0.999409i	$0.510948\pi$
$954$	0	0
$955$	0	0
$956$	722.179i	0.755418i
$957$	0	0
$958$	− 189.606i	− 0.197918i
$959$	− 73.5971i	− 0.0767435i
$960$	0	0
$961$	1397.32	1.45403
$962$	−158.652	−0.164919
$963$	0	0
$964$	−29.1798	−0.0302695
$965$	0	0
$966$	0	0
$967$	− 1095.49i	− 1.13287i	−0.824106	−	0.566436i	$-0.808322\pi$
$967$	− 1095.49i	− 1.13287i	0.824106	−	0.566436i	$-0.191678\pi$
$968$	220.091	0.227367
$969$	0	0
$970$	0	0
$971$	− 143.575i	− 0.147863i	−0.997263	−	0.0739313i	$-0.976445\pi$
$971$	− 143.575i	− 0.147863i	0.997263	−	0.0739313i	$-0.0235546\pi$
$972$	0	0
$973$	267.415i	0.274836i
$974$	− 392.952i	− 0.403441i
$975$	0	0
$976$	10.7642	0.0110289
$977$	1259.60	1.28925	0.644627	−	0.764498i	$-0.277013\pi$
$977$	1259.60	1.28925	0.644627	+	0.764498i	$0.277013\pi$
$978$	0	0
$979$	−1248.85	−1.27564
$980$	0	0
$981$	0	0
$982$	728.285i	0.741634i
$983$	−275.176	−0.279935	−0.139968	−	0.990156i	$-0.544700\pi$
$983$	−275.176	−0.279935	−0.139968	+	0.990156i	$0.544700\pi$
$984$	0	0
$985$	0	0
$986$	1852.47i	1.87878i
$987$	0	0
$988$	35.2991i	0.0357278i
$989$	− 487.131i	− 0.492549i
$990$	0	0
$991$	−1111.90	−1.12200	−0.560998	−	0.827817i	$-0.689583\pi$
$991$	−1111.90	−1.12200	−0.560998	+	0.827817i	$0.689583\pi$
$992$	−274.711	−0.276927
$993$	0	0
$994$	−152.108	−0.153026
$995$	0	0
$996$	0	0
$997$	− 1008.42i	− 1.01145i	−0.862693	−	0.505727i	$-0.831224\pi$
$997$	− 1008.42i	− 1.01145i	0.862693	−	0.505727i	$-0.168776\pi$
$998$	−203.720	−0.204129
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.f.449.8		16
3.2	odd	2	inner	3150.3.c.f.449.13		16
5.2	odd	4		3150.3.e.f.701.2		8
5.3	odd	4		630.3.e.b.71.6	yes	8
5.4	even	2	inner	3150.3.c.f.449.12		16
15.2	even	4		3150.3.e.f.701.5		8
15.8	even	4		630.3.e.b.71.4	✓	8
15.14	odd	2	inner	3150.3.c.f.449.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.3.e.b.71.4	✓	8	15.8	even	4
630.3.e.b.71.6	yes	8	5.3	odd	4
3150.3.c.f.449.1		16	15.14	odd	2	inner
3150.3.c.f.449.8		16	1.1	even	1	trivial
3150.3.c.f.449.12		16	5.4	even	2	inner
3150.3.c.f.449.13		16	3.2	odd	2	inner
3150.3.e.f.701.2		8	5.2	odd	4
3150.3.e.f.701.5		8	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:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +14.1001i q^{11} -1.78360i q^{13} -3.74166i q^{14} +4.00000 q^{16} +32.2729 q^{17} -9.89548 q^{19} -19.9406i q^{22} +9.58126 q^{23} +2.52238i q^{26} +5.29150i q^{28} -40.5881i q^{29} +48.5626 q^{31} -5.65685 q^{32} -45.6408 q^{34} +62.8976i q^{37} +13.9943 q^{38} -65.7349i q^{41} -50.8420i q^{43} +28.2003i q^{44} -13.5500 q^{46} -65.3405 q^{47} -7.00000 q^{49} -3.56719i q^{52} +64.9575 q^{53} -7.48331i q^{56} +57.4003i q^{58} -111.642i q^{59} +2.69105 q^{61} -68.6778 q^{62} +8.00000 q^{64} +72.1824i q^{67} +64.5458 q^{68} -40.6525i q^{71} +63.6965i q^{73} -88.9506i q^{74} -19.7910 q^{76} -37.3055 q^{77} +82.3035 q^{79} +92.9631i q^{82} -99.9021 q^{83} +71.9014i q^{86} -39.8812i q^{88} +88.5703i q^{89} +4.71895 q^{91} +19.1625 q^{92} +92.4054 q^{94} +117.848i q^{97} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 64 q^{16} + 160 q^{19} + 224 q^{31} - 192 q^{34} - 64 q^{46} - 112 q^{49} - 288 q^{61} + 128 q^{64} + 320 q^{76} + 128 q^{79} - 448 q^{91}+O(q^{100}) \) 16 * q + 32 * q^4 + 64 * q^16 + 160 * q^19 + 224 * q^31 - 192 * q^34 - 64 * q^46 - 112 * q^49 - 288 * q^61 + 128 * q^64 + 320 * q^76 + 128 * q^79 - 448 * q^91

Introduction

L-functions

Modular forms

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Representations

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Database

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