LMFDB - Embedded newform 675.4.b.a.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,4,Mod(649,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("675.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.b (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:	$39.8262892539$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	675.649
Dual form			675.4.b.a.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{2} -1.00000 q^{4} -25.0000i q^{7} +21.0000i q^{8} +O(q^{10})$	$q+3.00000i q^{2} -1.00000 q^{4} -25.0000i q^{7} +21.0000i q^{8} -15.0000 q^{11} -20.0000i q^{13} +75.0000 q^{14} -71.0000 q^{16} +72.0000i q^{17} -2.00000 q^{19} -45.0000i q^{22} -114.000i q^{23} +60.0000 q^{26} +25.0000i q^{28} -30.0000 q^{29} +101.000 q^{31} -45.0000i q^{32} -216.000 q^{34} -430.000i q^{37} -6.00000i q^{38} -30.0000 q^{41} -110.000i q^{43} +15.0000 q^{44} +342.000 q^{46} -330.000i q^{47} -282.000 q^{49} +20.0000i q^{52} -621.000i q^{53} +525.000 q^{56} -90.0000i q^{58} +660.000 q^{59} -376.000 q^{61} +303.000i q^{62} -433.000 q^{64} -250.000i q^{67} -72.0000i q^{68} -360.000 q^{71} -785.000i q^{73} +1290.00 q^{74} +2.00000 q^{76} +375.000i q^{77} -488.000 q^{79} -90.0000i q^{82} -489.000i q^{83} +330.000 q^{86} -315.000i q^{88} +450.000 q^{89} -500.000 q^{91} +114.000i q^{92} +990.000 q^{94} -1105.00i q^{97} -846.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 30 q^{11} + 150 q^{14} - 142 q^{16} - 4 q^{19} + 120 q^{26} - 60 q^{29} + 202 q^{31} - 432 q^{34} - 60 q^{41} + 30 q^{44} + 684 q^{46} - 564 q^{49} + 1050 q^{56} + 1320 q^{59} - 752 q^{61} - 866 q^{64} - 720 q^{71} + 2580 q^{74} + 4 q^{76} - 976 q^{79} + 660 q^{86} + 900 q^{89} - 1000 q^{91} + 1980 q^{94}+O(q^{100}) $ 2 * q - 2 * q^4 - 30 * q^11 + 150 * q^14 - 142 * q^16 - 4 * q^19 + 120 * q^26 - 60 * q^29 + 202 * q^31 - 432 * q^34 - 60 * q^41 + 30 * q^44 + 684 * q^46 - 564 * q^49 + 1050 * q^56 + 1320 * q^59 - 752 * q^61 - 866 * q^64 - 720 * q^71 + 2580 * q^74 + 4 * q^76 - 976 * q^79 + 660 * q^86 + 900 * q^89 - 1000 * q^91 + 1980 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$.

$n$	$326$	$352$
$\chi(n)$	$1$	$-1$

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 $

$2$	3.00000i	1.06066i	0.847791	+	0.530330i	$0.177932\pi$
$2$	3.00000i	1.06066i	−0.847791	+	0.530330i	$0.822068\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.125000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 25.0000i	− 1.34987i	−0.737876	−	0.674937i	$-0.764171\pi$
$7$	− 25.0000i	− 1.34987i	0.737876	−	0.674937i	$-0.235829\pi$
$8$	21.0000i	0.928078i
$9$	0	0
$10$	0	0
$11$	−15.0000	−0.411152	−0.205576	−	0.978641i	$-0.565907\pi$
$11$	−15.0000	−0.411152	−0.205576	+	0.978641i	$0.565907\pi$
$12$	0	0
$13$	− 20.0000i	− 0.426692i	−0.976977	−	0.213346i	$-0.931564\pi$
$13$	− 20.0000i	− 0.426692i	0.976977	−	0.213346i	$-0.0684362\pi$
$14$	75.0000	1.43176
$15$	0	0
$16$	−71.0000	−1.10938
$17$	72.0000i	1.02721i	0.858027	+	0.513605i	$0.171690\pi$
$17$	72.0000i	1.02721i	−0.858027	+	0.513605i	$0.828310\pi$
$18$	0	0
$19$	−2.00000	−0.0241490	−0.0120745	−	0.999927i	$-0.503844\pi$
$19$	−2.00000	−0.0241490	−0.0120745	+	0.999927i	$0.503844\pi$
$20$	0	0
$21$	0	0
$22$	− 45.0000i	− 0.436092i
$23$	− 114.000i	− 1.03351i	−0.856134	−	0.516753i	$-0.827141\pi$
$23$	− 114.000i	− 1.03351i	0.856134	−	0.516753i	$-0.172859\pi$
$24$	0	0
$25$	0	0
$26$	60.0000	0.452576
$27$	0	0
$28$	25.0000i	0.168734i
$29$	−30.0000	−0.192099	−0.0960493	−	0.995377i	$-0.530621\pi$
$29$	−30.0000	−0.192099	−0.0960493	+	0.995377i	$0.530621\pi$
$30$	0	0
$31$	101.000	0.585166	0.292583	−	0.956240i	$-0.405485\pi$
$31$	101.000	0.585166	0.292583	+	0.956240i	$0.405485\pi$
$32$	− 45.0000i	− 0.248592i
$33$	0	0
$34$	−216.000	−1.08952
$35$	0	0
$36$	0	0
$37$	− 430.000i	− 1.91058i	−0.295666	−	0.955291i	$-0.595542\pi$
$37$	− 430.000i	− 1.91058i	0.295666	−	0.955291i	$-0.404458\pi$
$38$	− 6.00000i	− 0.0256139i
$39$	0	0
$40$	0	0
$41$	−30.0000	−0.114273	−0.0571367	−	0.998366i	$-0.518197\pi$
$41$	−30.0000	−0.114273	−0.0571367	+	0.998366i	$0.518197\pi$
$42$	0	0
$43$	− 110.000i	− 0.390113i	−0.980792	−	0.195056i	$-0.937511\pi$
$43$	− 110.000i	− 0.390113i	0.980792	−	0.195056i	$-0.0624890\pi$
$44$	15.0000	0.0513940
$45$	0	0
$46$	342.000	1.09620
$47$	− 330.000i	− 1.02416i	−0.858938	−	0.512079i	$-0.828875\pi$
$47$	− 330.000i	− 1.02416i	0.858938	−	0.512079i	$-0.171125\pi$
$48$	0	0
$49$	−282.000	−0.822157
$50$	0	0
$51$	0	0
$52$	20.0000i	0.0533366i
$53$	− 621.000i	− 1.60945i	−0.593647	−	0.804726i	$-0.702312\pi$
$53$	− 621.000i	− 1.60945i	0.593647	−	0.804726i	$-0.297688\pi$
$54$	0	0
$55$	0	0
$56$	525.000	1.25279
$57$	0	0
$58$	− 90.0000i	− 0.203751i
$59$	660.000	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$59$	660.000	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$60$	0	0
$61$	−376.000	−0.789211	−0.394605	−	0.918851i	$-0.629119\pi$
$61$	−376.000	−0.789211	−0.394605	+	0.918851i	$0.629119\pi$
$62$	303.000i	0.620662i
$63$	0	0
$64$	−433.000	−0.845703
$65$	0	0
$66$	0	0
$67$	− 250.000i	− 0.455856i	−0.973678	−	0.227928i	$-0.926805\pi$
$67$	− 250.000i	− 0.455856i	0.973678	−	0.227928i	$-0.0731951\pi$
$68$	− 72.0000i	− 0.128401i
$69$	0	0
$70$	0	0
$71$	−360.000	−0.601748	−0.300874	−	0.953664i	$-0.597278\pi$
$71$	−360.000	−0.601748	−0.300874	+	0.953664i	$0.597278\pi$
$72$	0	0
$73$	− 785.000i	− 1.25859i	−0.777165	−	0.629297i	$-0.783343\pi$
$73$	− 785.000i	− 1.25859i	0.777165	−	0.629297i	$-0.216657\pi$
$74$	1290.00	2.02648
$75$	0	0
$76$	2.00000	0.00301863
$77$	375.000i	0.555003i
$78$	0	0
$79$	−488.000	−0.694991	−0.347496	−	0.937682i	$-0.612968\pi$
$79$	−488.000	−0.694991	−0.347496	+	0.937682i	$0.612968\pi$
$80$	0	0
$81$	0	0
$82$	− 90.0000i	− 0.121205i
$83$	− 489.000i	− 0.646683i	−0.946282	−	0.323342i	$-0.895194\pi$
$83$	− 489.000i	− 0.646683i	0.946282	−	0.323342i	$-0.104806\pi$
$84$	0	0
$85$	0	0
$86$	330.000	0.413777
$87$	0	0
$88$	− 315.000i	− 0.381581i
$89$	450.000	0.535954	0.267977	−	0.963425i	$-0.413645\pi$
$89$	450.000	0.535954	0.267977	+	0.963425i	$0.413645\pi$
$90$	0	0
$91$	−500.000	−0.575981
$92$	114.000i	0.129188i
$93$	0	0
$94$	990.000	1.08628
$95$	0	0
$96$	0	0
$97$	− 1105.00i	− 1.15666i	−0.815804	−	0.578329i	$-0.803705\pi$
$97$	− 1105.00i	− 1.15666i	0.815804	−	0.578329i	$-0.196295\pi$
$98$	− 846.000i	− 0.872030i
$99$	0	0
$100$	0	0
$101$	1425.00	1.40389	0.701945	−	0.712232i	$-0.252315\pi$
$101$	1425.00	1.40389	0.701945	+	0.712232i	$0.252315\pi$
$102$	0	0
$103$	1060.00i	1.01403i	0.861938	+	0.507014i	$0.169251\pi$
$103$	1060.00i	1.01403i	−0.861938	+	0.507014i	$0.830749\pi$
$104$	420.000	0.396004
$105$	0	0
$106$	1863.00	1.70708
$107$	1485.00i	1.34169i	0.741600	+	0.670843i	$0.234067\pi$
$107$	1485.00i	1.34169i	−0.741600	+	0.670843i	$0.765933\pi$
$108$	0	0
$109$	862.000	0.757474	0.378737	−	0.925504i	$-0.376359\pi$
$109$	862.000	0.757474	0.378737	+	0.925504i	$0.376359\pi$
$110$	0	0
$111$	0	0
$112$	1775.00i	1.49752i
$113$	− 690.000i	− 0.574422i	−0.957867	−	0.287211i	$-0.907272\pi$
$113$	− 690.000i	− 0.574422i	0.957867	−	0.287211i	$-0.0927282\pi$
$114$	0	0
$115$	0	0
$116$	30.0000	0.0240123
$117$	0	0
$118$	1980.00i	1.54469i
$119$	1800.00	1.38660
$120$	0	0
$121$	−1106.00	−0.830954
$122$	− 1128.00i	− 0.837085i
$123$	0	0
$124$	−101.000	−0.0731457
$125$	0	0
$126$	0	0
$127$	1865.00i	1.30309i	0.758611	+	0.651543i	$0.225878\pi$
$127$	1865.00i	1.30309i	−0.758611	+	0.651543i	$0.774122\pi$
$128$	− 1659.00i	− 1.14560i
$129$	0	0
$130$	0	0
$131$	−1155.00	−0.770327	−0.385163	−	0.922848i	$-0.625855\pi$
$131$	−1155.00	−0.770327	−0.385163	+	0.922848i	$0.625855\pi$
$132$	0	0
$133$	50.0000i	0.0325981i
$134$	750.000	0.483508
$135$	0	0
$136$	−1512.00	−0.953330
$137$	− 2778.00i	− 1.73241i	−0.499686	−	0.866206i	$-0.666551\pi$
$137$	− 2778.00i	− 1.73241i	0.499686	−	0.866206i	$-0.333449\pi$
$138$	0	0
$139$	1924.00	1.17404	0.587020	−	0.809572i	$-0.300301\pi$
$139$	1924.00	1.17404	0.587020	+	0.809572i	$0.300301\pi$
$140$	0	0
$141$	0	0
$142$	− 1080.00i	− 0.638251i
$143$	300.000i	0.175435i
$144$	0	0
$145$	0	0
$146$	2355.00	1.33494
$147$	0	0
$148$	430.000i	0.238823i
$149$	−1455.00	−0.799988	−0.399994	−	0.916518i	$-0.630988\pi$
$149$	−1455.00	−0.799988	−0.399994	+	0.916518i	$0.630988\pi$
$150$	0	0
$151$	−727.000	−0.391804	−0.195902	−	0.980623i	$-0.562763\pi$
$151$	−727.000	−0.391804	−0.195902	+	0.980623i	$0.562763\pi$
$152$	− 42.0000i	− 0.0224122i
$153$	0	0
$154$	−1125.00	−0.588669
$155$	0	0
$156$	0	0
$157$	3260.00i	1.65717i	0.559860	+	0.828587i	$0.310855\pi$
$157$	3260.00i	1.65717i	−0.559860	+	0.828587i	$0.689145\pi$
$158$	− 1464.00i	− 0.737149i
$159$	0	0
$160$	0	0
$161$	−2850.00	−1.39510
$162$	0	0
$163$	− 2540.00i	− 1.22054i	−0.792193	−	0.610270i	$-0.791061\pi$
$163$	− 2540.00i	− 1.22054i	0.792193	−	0.610270i	$-0.208939\pi$
$164$	30.0000	0.0142842
$165$	0	0
$166$	1467.00	0.685911
$167$	3498.00i	1.62086i	0.585837	+	0.810429i	$0.300766\pi$
$167$	3498.00i	1.62086i	−0.585837	+	0.810429i	$0.699234\pi$
$168$	0	0
$169$	1797.00	0.817934
$170$	0	0
$171$	0	0
$172$	110.000i	0.0487641i
$173$	1149.00i	0.504953i	0.967603	+	0.252476i	$0.0812450\pi$
$173$	1149.00i	0.504953i	−0.967603	+	0.252476i	$0.918755\pi$
$174$	0	0
$175$	0	0
$176$	1065.00	0.456122
$177$	0	0
$178$	1350.00i	0.568465i
$179$	−315.000	−0.131532	−0.0657659	−	0.997835i	$-0.520949\pi$
$179$	−315.000	−0.131532	−0.0657659	+	0.997835i	$0.520949\pi$
$180$	0	0
$181$	1136.00	0.466509	0.233255	−	0.972416i	$-0.425062\pi$
$181$	1136.00	0.466509	0.233255	+	0.972416i	$0.425062\pi$
$182$	− 1500.00i	− 0.610920i
$183$	0	0
$184$	2394.00	0.959174
$185$	0	0
$186$	0	0
$187$	− 1080.00i	− 0.422339i
$188$	330.000i	0.128020i
$189$	0	0
$190$	0	0
$191$	2460.00	0.931934	0.465967	−	0.884802i	$-0.345707\pi$
$191$	2460.00	0.931934	0.465967	+	0.884802i	$0.345707\pi$
$192$	0	0
$193$	− 965.000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 965.000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	3315.00	1.22682
$195$	0	0
$196$	282.000	0.102770
$197$	2493.00i	0.901619i	0.892620	+	0.450809i	$0.148865\pi$
$197$	2493.00i	0.901619i	−0.892620	+	0.450809i	$0.851135\pi$
$198$	0	0
$199$	511.000	0.182029	0.0910146	−	0.995850i	$-0.470989\pi$
$199$	511.000	0.182029	0.0910146	+	0.995850i	$0.470989\pi$
$200$	0	0
$201$	0	0
$202$	4275.00i	1.48905i
$203$	750.000i	0.259309i
$204$	0	0
$205$	0	0
$206$	−3180.00	−1.07554
$207$	0	0
$208$	1420.00i	0.473362i
$209$	30.0000	0.00992892
$210$	0	0
$211$	−2086.00	−0.680598	−0.340299	−	0.940317i	$-0.610528\pi$
$211$	−2086.00	−0.680598	−0.340299	+	0.940317i	$0.610528\pi$
$212$	621.000i	0.201181i
$213$	0	0
$214$	−4455.00	−1.42307
$215$	0	0
$216$	0	0
$217$	− 2525.00i	− 0.789899i
$218$	2586.00i	0.803422i
$219$	0	0
$220$	0	0
$221$	1440.00	0.438303
$222$	0	0
$223$	− 5240.00i	− 1.57353i	−0.617255	−	0.786763i	$-0.711755\pi$
$223$	− 5240.00i	− 1.57353i	0.617255	−	0.786763i	$-0.288245\pi$
$224$	−1125.00	−0.335568
$225$	0	0
$226$	2070.00	0.609267
$227$	2388.00i	0.698225i	0.937081	+	0.349113i	$0.113517\pi$
$227$	2388.00i	0.698225i	−0.937081	+	0.349113i	$0.886483\pi$
$228$	0	0
$229$	−182.000	−0.0525192	−0.0262596	−	0.999655i	$-0.508360\pi$
$229$	−182.000	−0.0525192	−0.0262596	+	0.999655i	$0.508360\pi$
$230$	0	0
$231$	0	0
$232$	− 630.000i	− 0.178282i
$233$	− 450.000i	− 0.126526i	−0.997997	−	0.0632628i	$-0.979849\pi$
$233$	− 450.000i	− 0.126526i	0.997997	−	0.0632628i	$-0.0201506\pi$
$234$	0	0
$235$	0	0
$236$	−660.000	−0.182044
$237$	0	0
$238$	5400.00i	1.47071i
$239$	−5190.00	−1.40466	−0.702329	−	0.711853i	$-0.747856\pi$
$239$	−5190.00	−1.40466	−0.702329	+	0.711853i	$0.747856\pi$
$240$	0	0
$241$	−2266.00	−0.605668	−0.302834	−	0.953043i	$-0.597933\pi$
$241$	−2266.00	−0.605668	−0.302834	+	0.953043i	$0.597933\pi$
$242$	− 3318.00i	− 0.881360i
$243$	0	0
$244$	376.000	0.0986514
$245$	0	0
$246$	0	0
$247$	40.0000i	0.0103042i
$248$	2121.00i	0.543079i
$249$	0	0
$250$	0	0
$251$	−2880.00	−0.724239	−0.362119	−	0.932132i	$-0.617947\pi$
$251$	−2880.00	−0.724239	−0.362119	+	0.932132i	$0.617947\pi$
$252$	0	0
$253$	1710.00i	0.424928i
$254$	−5595.00	−1.38213
$255$	0	0
$256$	1513.00	0.369385
$257$	− 4188.00i	− 1.01650i	−0.861210	−	0.508250i	$-0.830293\pi$
$257$	− 4188.00i	− 1.01650i	0.861210	−	0.508250i	$-0.169707\pi$
$258$	0	0
$259$	−10750.0	−2.57904
$260$	0	0
$261$	0	0
$262$	− 3465.00i	− 0.817055i
$263$	3030.00i	0.710410i	0.934788	+	0.355205i	$0.115589\pi$
$263$	3030.00i	0.710410i	−0.934788	+	0.355205i	$0.884411\pi$
$264$	0	0
$265$	0	0
$266$	−150.000	−0.0345755
$267$	0	0
$268$	250.000i	0.0569820i
$269$	−3510.00	−0.795571	−0.397785	−	0.917479i	$-0.630221\pi$
$269$	−3510.00	−0.795571	−0.397785	+	0.917479i	$0.630221\pi$
$270$	0	0
$271$	2999.00	0.672237	0.336119	−	0.941820i	$-0.390886\pi$
$271$	2999.00	0.672237	0.336119	+	0.941820i	$0.390886\pi$
$272$	− 5112.00i	− 1.13956i
$273$	0	0
$274$	8334.00	1.83750
$275$	0	0
$276$	0	0
$277$	− 7720.00i	− 1.67455i	−0.546783	−	0.837274i	$-0.684148\pi$
$277$	− 7720.00i	− 1.67455i	0.546783	−	0.837274i	$-0.315852\pi$
$278$	5772.00i	1.24526i
$279$	0	0
$280$	0	0
$281$	−7440.00	−1.57948	−0.789739	−	0.613443i	$-0.789784\pi$
$281$	−7440.00	−1.57948	−0.789739	+	0.613443i	$0.789784\pi$
$282$	0	0
$283$	− 830.000i	− 0.174341i	−0.996193	−	0.0871703i	$-0.972218\pi$
$283$	− 830.000i	− 0.174341i	0.996193	−	0.0871703i	$-0.0277824\pi$
$284$	360.000	0.0752186
$285$	0	0
$286$	−900.000	−0.186077
$287$	750.000i	0.154255i
$288$	0	0
$289$	−271.000	−0.0551598
$290$	0	0
$291$	0	0
$292$	785.000i	0.157324i
$293$	− 546.000i	− 0.108866i	−0.998517	−	0.0544329i	$-0.982665\pi$
$293$	− 546.000i	− 0.108866i	0.998517	−	0.0544329i	$-0.0173351\pi$
$294$	0	0
$295$	0	0
$296$	9030.00	1.77317
$297$	0	0
$298$	− 4365.00i	− 0.848516i
$299$	−2280.00	−0.440989
$300$	0	0
$301$	−2750.00	−0.526603
$302$	− 2181.00i	− 0.415571i
$303$	0	0
$304$	142.000	0.0267903
$305$	0	0
$306$	0	0
$307$	− 5560.00i	− 1.03364i	−0.856096	−	0.516818i	$-0.827117\pi$
$307$	− 5560.00i	− 1.03364i	0.856096	−	0.516818i	$-0.172883\pi$
$308$	− 375.000i	− 0.0693754i
$309$	0	0
$310$	0	0
$311$	−8670.00	−1.58081	−0.790403	−	0.612587i	$-0.790129\pi$
$311$	−8670.00	−1.58081	−0.790403	+	0.612587i	$0.790129\pi$
$312$	0	0
$313$	− 4565.00i	− 0.824374i	−0.911099	−	0.412187i	$-0.864765\pi$
$313$	− 4565.00i	− 0.824374i	0.911099	−	0.412187i	$-0.135235\pi$
$314$	−9780.00	−1.75770
$315$	0	0
$316$	488.000	0.0868739
$317$	4233.00i	0.749997i	0.927025	+	0.374998i	$0.122357\pi$
$317$	4233.00i	0.749997i	−0.927025	+	0.374998i	$0.877643\pi$
$318$	0	0
$319$	450.000	0.0789817
$320$	0	0
$321$	0	0
$322$	− 8550.00i	− 1.47973i
$323$	− 144.000i	− 0.0248061i
$324$	0	0
$325$	0	0
$326$	7620.00	1.29458
$327$	0	0
$328$	− 630.000i	− 0.106055i
$329$	−8250.00	−1.38248
$330$	0	0
$331$	542.000	0.0900031	0.0450015	−	0.998987i	$-0.485671\pi$
$331$	542.000	0.0900031	0.0450015	+	0.998987i	$0.485671\pi$
$332$	489.000i	0.0808354i
$333$	0	0
$334$	−10494.0	−1.71918
$335$	0	0
$336$	0	0
$337$	5690.00i	0.919745i	0.887985	+	0.459872i	$0.152105\pi$
$337$	5690.00i	0.919745i	−0.887985	+	0.459872i	$0.847895\pi$
$338$	5391.00i	0.867550i
$339$	0	0
$340$	0	0
$341$	−1515.00	−0.240592
$342$	0	0
$343$	− 1525.00i	− 0.240065i
$344$	2310.00	0.362055
$345$	0	0
$346$	−3447.00	−0.535583
$347$	5055.00i	0.782036i	0.920383	+	0.391018i	$0.127877\pi$
$347$	5055.00i	0.782036i	−0.920383	+	0.391018i	$0.872123\pi$
$348$	0	0
$349$	−1622.00	−0.248778	−0.124389	−	0.992234i	$-0.539697\pi$
$349$	−1622.00	−0.248778	−0.124389	+	0.992234i	$0.539697\pi$
$350$	0	0
$351$	0	0
$352$	675.000i	0.102209i
$353$	− 30.0000i	− 0.00452334i	−0.999997	−	0.00226167i	$-0.999280\pi$
$353$	− 30.0000i	− 0.00452334i	0.999997	−	0.00226167i	$-0.000719912\pi$
$354$	0	0
$355$	0	0
$356$	−450.000	−0.0669942
$357$	0	0
$358$	− 945.000i	− 0.139511i
$359$	7470.00	1.09819	0.549097	−	0.835759i	$-0.314972\pi$
$359$	7470.00	1.09819	0.549097	+	0.835759i	$0.314972\pi$
$360$	0	0
$361$	−6855.00	−0.999417
$362$	3408.00i	0.494808i
$363$	0	0
$364$	500.000	0.0719976
$365$	0	0
$366$	0	0
$367$	− 1375.00i	− 0.195571i	−0.995208	−	0.0977853i	$-0.968824\pi$
$367$	− 1375.00i	− 0.195571i	0.995208	−	0.0977853i	$-0.0311758\pi$
$368$	8094.00i	1.14655i
$369$	0	0
$370$	0	0
$371$	−15525.0	−2.17255
$372$	0	0
$373$	4840.00i	0.671865i	0.941886	+	0.335933i	$0.109051\pi$
$373$	4840.00i	0.671865i	−0.941886	+	0.335933i	$0.890949\pi$
$374$	3240.00	0.447958
$375$	0	0
$376$	6930.00	0.950499
$377$	600.000i	0.0819670i
$378$	0	0
$379$	−1892.00	−0.256426	−0.128213	−	0.991747i	$-0.540924\pi$
$379$	−1892.00	−0.256426	−0.128213	+	0.991747i	$0.540924\pi$
$380$	0	0
$381$	0	0
$382$	7380.00i	0.988465i
$383$	10704.0i	1.42806i	0.700113	+	0.714032i	$0.253133\pi$
$383$	10704.0i	1.42806i	−0.700113	+	0.714032i	$0.746867\pi$
$384$	0	0
$385$	0	0
$386$	2895.00	0.381740
$387$	0	0
$388$	1105.00i	0.144582i
$389$	−7815.00	−1.01860	−0.509301	−	0.860588i	$-0.670096\pi$
$389$	−7815.00	−1.01860	−0.509301	+	0.860588i	$0.670096\pi$
$390$	0	0
$391$	8208.00	1.06163
$392$	− 5922.00i	− 0.763026i
$393$	0	0
$394$	−7479.00	−0.956311
$395$	0	0
$396$	0	0
$397$	4700.00i	0.594172i	0.954851	+	0.297086i	$0.0960148\pi$
$397$	4700.00i	0.594172i	−0.954851	+	0.297086i	$0.903985\pi$
$398$	1533.00i	0.193071i
$399$	0	0
$400$	0	0
$401$	−2100.00	−0.261519	−0.130759	−	0.991414i	$-0.541742\pi$
$401$	−2100.00	−0.261519	−0.130759	+	0.991414i	$0.541742\pi$
$402$	0	0
$403$	− 2020.00i	− 0.249686i
$404$	−1425.00	−0.175486
$405$	0	0
$406$	−2250.00	−0.275038
$407$	6450.00i	0.785540i
$408$	0	0
$409$	10753.0	1.30000	0.650002	−	0.759933i	$-0.274768\pi$
$409$	10753.0	1.30000	0.650002	+	0.759933i	$0.274768\pi$
$410$	0	0
$411$	0	0
$412$	− 1060.00i	− 0.126754i
$413$	− 16500.0i	− 1.96589i
$414$	0	0
$415$	0	0
$416$	−900.000	−0.106072
$417$	0	0
$418$	90.0000i	0.0105312i
$419$	−2940.00	−0.342789	−0.171394	−	0.985203i	$-0.554827\pi$
$419$	−2940.00	−0.342789	−0.171394	+	0.985203i	$0.554827\pi$
$420$	0	0
$421$	8696.00	1.00669	0.503346	−	0.864085i	$-0.332102\pi$
$421$	8696.00	1.00669	0.503346	+	0.864085i	$0.332102\pi$
$422$	− 6258.00i	− 0.721883i
$423$	0	0
$424$	13041.0	1.49370
$425$	0	0
$426$	0	0
$427$	9400.00i	1.06533i
$428$	− 1485.00i	− 0.167711i
$429$	0	0
$430$	0	0
$431$	8370.00	0.935426	0.467713	−	0.883880i	$-0.345078\pi$
$431$	8370.00	0.935426	0.467713	+	0.883880i	$0.345078\pi$
$432$	0	0
$433$	5155.00i	0.572133i	0.958210	+	0.286066i	$0.0923478\pi$
$433$	5155.00i	0.572133i	−0.958210	+	0.286066i	$0.907652\pi$
$434$	7575.00	0.837815
$435$	0	0
$436$	−862.000	−0.0946842
$437$	228.000i	0.0249582i
$438$	0	0
$439$	10987.0	1.19449	0.597245	−	0.802059i	$-0.296262\pi$
$439$	10987.0	1.19449	0.597245	+	0.802059i	$0.296262\pi$
$440$	0	0
$441$	0	0
$442$	4320.00i	0.464890i
$443$	− 1956.00i	− 0.209780i	−0.994484	−	0.104890i	$-0.966551\pi$
$443$	− 1956.00i	− 0.209780i	0.994484	−	0.104890i	$-0.0334490\pi$
$444$	0	0
$445$	0	0
$446$	15720.0	1.66898
$447$	0	0
$448$	10825.0i	1.14159i
$449$	8730.00	0.917582	0.458791	−	0.888544i	$-0.348283\pi$
$449$	8730.00	0.917582	0.458791	+	0.888544i	$0.348283\pi$
$450$	0	0
$451$	450.000	0.0469838
$452$	690.000i	0.0718028i
$453$	0	0
$454$	−7164.00	−0.740580
$455$	0	0
$456$	0	0
$457$	− 8665.00i	− 0.886940i	−0.896289	−	0.443470i	$-0.853747\pi$
$457$	− 8665.00i	− 0.886940i	0.896289	−	0.443470i	$-0.146253\pi$
$458$	− 546.000i	− 0.0557050i
$459$	0	0
$460$	0	0
$461$	−9825.00	−0.992616	−0.496308	−	0.868147i	$-0.665311\pi$
$461$	−9825.00	−0.992616	−0.496308	+	0.868147i	$0.665311\pi$
$462$	0	0
$463$	5245.00i	0.526470i	0.964732	+	0.263235i	$0.0847895\pi$
$463$	5245.00i	0.526470i	−0.964732	+	0.263235i	$0.915210\pi$
$464$	2130.00	0.213109
$465$	0	0
$466$	1350.00	0.134201
$467$	− 11007.0i	− 1.09067i	−0.838218	−	0.545335i	$-0.816402\pi$
$467$	− 11007.0i	− 1.09067i	0.838218	−	0.545335i	$-0.183598\pi$
$468$	0	0
$469$	−6250.00	−0.615348
$470$	0	0
$471$	0	0
$472$	13860.0i	1.35161i
$473$	1650.00i	0.160396i
$474$	0	0
$475$	0	0
$476$	−1800.00	−0.173325
$477$	0	0
$478$	− 15570.0i	− 1.48986i
$479$	−16950.0	−1.61684	−0.808419	−	0.588608i	$-0.799676\pi$
$479$	−16950.0	−1.61684	−0.808419	+	0.588608i	$0.799676\pi$
$480$	0	0
$481$	−8600.00	−0.815231
$482$	− 6798.00i	− 0.642408i
$483$	0	0
$484$	1106.00	0.103869
$485$	0	0
$486$	0	0
$487$	10640.0i	0.990030i	0.868885	+	0.495015i	$0.164837\pi$
$487$	10640.0i	0.990030i	−0.868885	+	0.495015i	$0.835163\pi$
$488$	− 7896.00i	− 0.732449i
$489$	0	0
$490$	0	0
$491$	1635.00	0.150278	0.0751390	−	0.997173i	$-0.476060\pi$
$491$	1635.00	0.150278	0.0751390	+	0.997173i	$0.476060\pi$
$492$	0	0
$493$	− 2160.00i	− 0.197326i
$494$	−120.000	−0.0109293
$495$	0	0
$496$	−7171.00	−0.649168
$497$	9000.00i	0.812284i
$498$	0	0
$499$	15802.0	1.41762	0.708812	−	0.705397i	$-0.249231\pi$
$499$	15802.0	1.41762	0.708812	+	0.705397i	$0.249231\pi$
$500$	0	0
$501$	0	0
$502$	− 8640.00i	− 0.768171i
$503$	7866.00i	0.697272i	0.937258	+	0.348636i	$0.113355\pi$
$503$	7866.00i	0.697272i	−0.937258	+	0.348636i	$0.886645\pi$
$504$	0	0
$505$	0	0
$506$	−5130.00	−0.450704
$507$	0	0
$508$	− 1865.00i	− 0.162886i
$509$	11955.0	1.04105	0.520527	−	0.853845i	$-0.325736\pi$
$509$	11955.0	1.04105	0.520527	+	0.853845i	$0.325736\pi$
$510$	0	0
$511$	−19625.0	−1.69894
$512$	− 8733.00i	− 0.753804i
$513$	0	0
$514$	12564.0	1.07816
$515$	0	0
$516$	0	0
$517$	4950.00i	0.421085i
$518$	− 32250.0i	− 2.73549i
$519$	0	0
$520$	0	0
$521$	19260.0	1.61957	0.809785	−	0.586727i	$-0.199584\pi$
$521$	19260.0	1.61957	0.809785	+	0.586727i	$0.199584\pi$
$522$	0	0
$523$	18520.0i	1.54842i	0.632930	+	0.774209i	$0.281852\pi$
$523$	18520.0i	1.54842i	−0.632930	+	0.774209i	$0.718148\pi$
$524$	1155.00	0.0962909
$525$	0	0
$526$	−9090.00	−0.753503
$527$	7272.00i	0.601088i
$528$	0	0
$529$	−829.000	−0.0681351
$530$	0	0
$531$	0	0
$532$	− 50.0000i	− 0.00407476i
$533$	600.000i	0.0487596i
$534$	0	0
$535$	0	0
$536$	5250.00	0.423070
$537$	0	0
$538$	− 10530.0i	− 0.843830i
$539$	4230.00	0.338032
$540$	0	0
$541$	8372.00	0.665324	0.332662	−	0.943046i	$-0.392053\pi$
$541$	8372.00	0.665324	0.332662	+	0.943046i	$0.392053\pi$
$542$	8997.00i	0.713015i
$543$	0	0
$544$	3240.00	0.255356
$545$	0	0
$546$	0	0
$547$	17120.0i	1.33821i	0.743170	+	0.669103i	$0.233321\pi$
$547$	17120.0i	1.33821i	−0.743170	+	0.669103i	$0.766679\pi$
$548$	2778.00i	0.216552i
$549$	0	0
$550$	0	0
$551$	60.0000	0.00463899
$552$	0	0
$553$	12200.0i	0.938150i
$554$	23160.0	1.77613
$555$	0	0
$556$	−1924.00	−0.146755
$557$	10575.0i	0.804447i	0.915541	+	0.402224i	$0.131763\pi$
$557$	10575.0i	0.804447i	−0.915541	+	0.402224i	$0.868237\pi$
$558$	0	0
$559$	−2200.00	−0.166458
$560$	0	0
$561$	0	0
$562$	− 22320.0i	− 1.67529i
$563$	− 10455.0i	− 0.782639i	−0.920255	−	0.391319i	$-0.872019\pi$
$563$	− 10455.0i	− 0.782639i	0.920255	−	0.391319i	$-0.127981\pi$
$564$	0	0
$565$	0	0
$566$	2490.00	0.184916
$567$	0	0
$568$	− 7560.00i	− 0.558469i
$569$	24540.0	1.80803	0.904016	−	0.427498i	$-0.140605\pi$
$569$	24540.0	1.80803	0.904016	+	0.427498i	$0.140605\pi$
$570$	0	0
$571$	24644.0	1.80616	0.903082	−	0.429469i	$-0.141299\pi$
$571$	24644.0	1.80616	0.903082	+	0.429469i	$0.141299\pi$
$572$	− 300.000i	− 0.0219294i
$573$	0	0
$574$	−2250.00	−0.163612
$575$	0	0
$576$	0	0
$577$	− 9610.00i	− 0.693361i	−0.937983	−	0.346681i	$-0.887309\pi$
$577$	− 9610.00i	− 0.693361i	0.937983	−	0.346681i	$-0.112691\pi$
$578$	− 813.000i	− 0.0585058i
$579$	0	0
$580$	0	0
$581$	−12225.0	−0.872941
$582$	0	0
$583$	9315.00i	0.661729i
$584$	16485.0	1.16807
$585$	0	0
$586$	1638.00	0.115470
$587$	4017.00i	0.282452i	0.989977	+	0.141226i	$0.0451044\pi$
$587$	4017.00i	0.282452i	−0.989977	+	0.141226i	$0.954896\pi$
$588$	0	0
$589$	−202.000	−0.0141312
$590$	0	0
$591$	0	0
$592$	30530.0i	2.11955i
$593$	− 594.000i	− 0.0411343i	−0.999788	−	0.0205672i	$-0.993453\pi$
$593$	− 594.000i	− 0.0411343i	0.999788	−	0.0205672i	$-0.00654719\pi$
$594$	0	0
$595$	0	0
$596$	1455.00	0.0999985
$597$	0	0
$598$	− 6840.00i	− 0.467740i
$599$	−8790.00	−0.599582	−0.299791	−	0.954005i	$-0.596917\pi$
$599$	−8790.00	−0.599582	−0.299791	+	0.954005i	$0.596917\pi$
$600$	0	0
$601$	9371.00	0.636025	0.318013	−	0.948087i	$-0.396985\pi$
$601$	9371.00	0.636025	0.318013	+	0.948087i	$0.396985\pi$
$602$	− 8250.00i	− 0.558546i
$603$	0	0
$604$	727.000	0.0489755
$605$	0	0
$606$	0	0
$607$	− 14560.0i	− 0.973595i	−0.873515	−	0.486798i	$-0.838165\pi$
$607$	− 14560.0i	− 0.973595i	0.873515	−	0.486798i	$-0.161835\pi$
$608$	90.0000i	0.00600326i
$609$	0	0
$610$	0	0
$611$	−6600.00	−0.437001
$612$	0	0
$613$	18250.0i	1.20246i	0.799074	+	0.601232i	$0.205323\pi$
$613$	18250.0i	1.20246i	−0.799074	+	0.601232i	$0.794677\pi$
$614$	16680.0	1.09634
$615$	0	0
$616$	−7875.00	−0.515086
$617$	19662.0i	1.28292i	0.767156	+	0.641461i	$0.221671\pi$
$617$	19662.0i	1.28292i	−0.767156	+	0.641461i	$0.778329\pi$
$618$	0	0
$619$	−12044.0	−0.782050	−0.391025	−	0.920380i	$-0.627879\pi$
$619$	−12044.0	−0.782050	−0.391025	+	0.920380i	$0.627879\pi$
$620$	0	0
$621$	0	0
$622$	− 26010.0i	− 1.67670i
$623$	− 11250.0i	− 0.723470i
$624$	0	0
$625$	0	0
$626$	13695.0	0.874381
$627$	0	0
$628$	− 3260.00i	− 0.207147i
$629$	30960.0	1.96257
$630$	0	0
$631$	14879.0	0.938706	0.469353	−	0.883011i	$-0.344487\pi$
$631$	14879.0	0.938706	0.469353	+	0.883011i	$0.344487\pi$
$632$	− 10248.0i	− 0.645006i
$633$	0	0
$634$	−12699.0	−0.795492
$635$	0	0
$636$	0	0
$637$	5640.00i	0.350808i
$638$	1350.00i	0.0837727i
$639$	0	0
$640$	0	0
$641$	8850.00	0.545326	0.272663	−	0.962110i	$-0.412096\pi$
$641$	8850.00	0.545326	0.272663	+	0.962110i	$0.412096\pi$
$642$	0	0
$643$	− 18380.0i	− 1.12727i	−0.826023	−	0.563636i	$-0.809402\pi$
$643$	− 18380.0i	− 1.12727i	0.826023	−	0.563636i	$-0.190598\pi$
$644$	2850.00	0.174388
$645$	0	0
$646$	432.000	0.0263109
$647$	3888.00i	0.236249i	0.992999	+	0.118124i	$0.0376882\pi$
$647$	3888.00i	0.236249i	−0.992999	+	0.118124i	$0.962312\pi$
$648$	0	0
$649$	−9900.00	−0.598781
$650$	0	0
$651$	0	0
$652$	2540.00i	0.152568i
$653$	6789.00i	0.406852i	0.979090	+	0.203426i	$0.0652076\pi$
$653$	6789.00i	0.406852i	−0.979090	+	0.203426i	$0.934792\pi$
$654$	0	0
$655$	0	0
$656$	2130.00	0.126772
$657$	0	0
$658$	− 24750.0i	− 1.46635i
$659$	−28335.0	−1.67492	−0.837462	−	0.546496i	$-0.815962\pi$
$659$	−28335.0	−1.67492	−0.837462	+	0.546496i	$0.815962\pi$
$660$	0	0
$661$	−6082.00	−0.357886	−0.178943	−	0.983859i	$-0.557268\pi$
$661$	−6082.00	−0.357886	−0.178943	+	0.983859i	$0.557268\pi$
$662$	1626.00i	0.0954627i
$663$	0	0
$664$	10269.0	0.600172
$665$	0	0
$666$	0	0
$667$	3420.00i	0.198535i
$668$	− 3498.00i	− 0.202607i
$669$	0	0
$670$	0	0
$671$	5640.00	0.324486
$672$	0	0
$673$	− 9965.00i	− 0.570762i	−0.958414	−	0.285381i	$-0.907880\pi$
$673$	− 9965.00i	− 0.570762i	0.958414	−	0.285381i	$-0.0921201\pi$
$674$	−17070.0	−0.975537
$675$	0	0
$676$	−1797.00	−0.102242
$677$	8130.00i	0.461538i	0.973009	+	0.230769i	$0.0741242\pi$
$677$	8130.00i	0.461538i	−0.973009	+	0.230769i	$0.925876\pi$
$678$	0	0
$679$	−27625.0	−1.56134
$680$	0	0
$681$	0	0
$682$	− 4545.00i	− 0.255186i
$683$	− 33516.0i	− 1.87768i	−0.344356	−	0.938839i	$-0.611903\pi$
$683$	− 33516.0i	− 1.87768i	0.344356	−	0.938839i	$-0.388097\pi$
$684$	0	0
$685$	0	0
$686$	4575.00	0.254627
$687$	0	0
$688$	7810.00i	0.432781i
$689$	−12420.0	−0.686741
$690$	0	0
$691$	−22084.0	−1.21580	−0.607898	−	0.794015i	$-0.707987\pi$
$691$	−22084.0	−1.21580	−0.607898	+	0.794015i	$0.707987\pi$
$692$	− 1149.00i	− 0.0631191i
$693$	0	0
$694$	−15165.0	−0.829475
$695$	0	0
$696$	0	0
$697$	− 2160.00i	− 0.117383i
$698$	− 4866.00i	− 0.263869i
$699$	0	0
$700$	0	0
$701$	−10395.0	−0.560077	−0.280038	−	0.959989i	$-0.590347\pi$
$701$	−10395.0	−0.560077	−0.280038	+	0.959989i	$0.590347\pi$
$702$	0	0
$703$	860.000i	0.0461387i
$704$	6495.00	0.347712
$705$	0	0
$706$	90.0000	0.00479773
$707$	− 35625.0i	− 1.89507i
$708$	0	0
$709$	4804.00	0.254468	0.127234	−	0.991873i	$-0.459390\pi$
$709$	4804.00	0.254468	0.127234	+	0.991873i	$0.459390\pi$
$710$	0	0
$711$	0	0
$712$	9450.00i	0.497407i
$713$	− 11514.0i	− 0.604772i
$714$	0	0
$715$	0	0
$716$	315.000	0.0164415
$717$	0	0
$718$	22410.0i	1.16481i
$719$	−10980.0	−0.569520	−0.284760	−	0.958599i	$-0.591914\pi$
$719$	−10980.0	−0.569520	−0.284760	+	0.958599i	$0.591914\pi$
$720$	0	0
$721$	26500.0	1.36881
$722$	− 20565.0i	− 1.06004i
$723$	0	0
$724$	−1136.00	−0.0583137
$725$	0	0
$726$	0	0
$727$	− 25945.0i	− 1.32359i	−0.749687	−	0.661793i	$-0.769796\pi$
$727$	− 25945.0i	− 1.32359i	0.749687	−	0.661793i	$-0.230204\pi$
$728$	− 10500.0i	− 0.534555i
$729$	0	0
$730$	0	0
$731$	7920.00	0.400727
$732$	0	0
$733$	− 18650.0i	− 0.939773i	−0.882727	−	0.469886i	$-0.844295\pi$
$733$	− 18650.0i	− 0.939773i	0.882727	−	0.469886i	$-0.155705\pi$
$734$	4125.00	0.207434
$735$	0	0
$736$	−5130.00	−0.256922
$737$	3750.00i	0.187426i
$738$	0	0
$739$	5128.00	0.255259	0.127630	−	0.991822i	$-0.459263\pi$
$739$	5128.00	0.255259	0.127630	+	0.991822i	$0.459263\pi$
$740$	0	0
$741$	0	0
$742$	− 46575.0i	− 2.30434i
$743$	32700.0i	1.61460i	0.590142	+	0.807299i	$0.299072\pi$
$743$	32700.0i	1.61460i	−0.590142	+	0.807299i	$0.700928\pi$
$744$	0	0
$745$	0	0
$746$	−14520.0	−0.712621
$747$	0	0
$748$	1080.00i	0.0527924i
$749$	37125.0	1.81111
$750$	0	0
$751$	21161.0	1.02820	0.514098	−	0.857731i	$-0.328127\pi$
$751$	21161.0	1.02820	0.514098	+	0.857731i	$0.328127\pi$
$752$	23430.0i	1.13618i
$753$	0	0
$754$	−1800.00	−0.0869392
$755$	0	0
$756$	0	0
$757$	7130.00i	0.342331i	0.985242	+	0.171165i	$0.0547532\pi$
$757$	7130.00i	0.342331i	−0.985242	+	0.171165i	$0.945247\pi$
$758$	− 5676.00i	− 0.271981i
$759$	0	0
$760$	0	0
$761$	−3360.00	−0.160052	−0.0800262	−	0.996793i	$-0.525500\pi$
$761$	−3360.00	−0.160052	−0.0800262	+	0.996793i	$0.525500\pi$
$762$	0	0
$763$	− 21550.0i	− 1.02249i
$764$	−2460.00	−0.116492
$765$	0	0
$766$	−32112.0	−1.51469
$767$	− 13200.0i	− 0.621414i
$768$	0	0
$769$	−33473.0	−1.56966	−0.784829	−	0.619712i	$-0.787249\pi$
$769$	−33473.0	−1.56966	−0.784829	+	0.619712i	$0.787249\pi$
$770$	0	0
$771$	0	0
$772$	965.000i	0.0449885i
$773$	3546.00i	0.164995i	0.996591	+	0.0824973i	$0.0262896\pi$
$773$	3546.00i	0.164995i	−0.996591	+	0.0824973i	$0.973710\pi$
$774$	0	0
$775$	0	0
$776$	23205.0	1.07347
$777$	0	0
$778$	− 23445.0i	− 1.08039i
$779$	60.0000	0.00275959
$780$	0	0
$781$	5400.00	0.247410
$782$	24624.0i	1.12603i
$783$	0	0
$784$	20022.0	0.912081
$785$	0	0
$786$	0	0
$787$	− 31840.0i	− 1.44215i	−0.692856	−	0.721076i	$-0.743648\pi$
$787$	− 31840.0i	− 1.44215i	0.692856	−	0.721076i	$-0.256352\pi$
$788$	− 2493.00i	− 0.112702i
$789$	0	0
$790$	0	0
$791$	−17250.0	−0.775397
$792$	0	0
$793$	7520.00i	0.336750i
$794$	−14100.0	−0.630214
$795$	0	0
$796$	−511.000	−0.0227537
$797$	− 15717.0i	− 0.698525i	−0.937025	−	0.349263i	$-0.886432\pi$
$797$	− 15717.0i	− 0.698525i	0.937025	−	0.349263i	$-0.113568\pi$
$798$	0	0
$799$	23760.0	1.05203
$800$	0	0
$801$	0	0
$802$	− 6300.00i	− 0.277382i
$803$	11775.0i	0.517473i
$804$	0	0
$805$	0	0
$806$	6060.00	0.264832
$807$	0	0
$808$	29925.0i	1.30292i
$809$	−10530.0	−0.457621	−0.228810	−	0.973471i	$-0.573484\pi$
$809$	−10530.0	−0.457621	−0.228810	+	0.973471i	$0.573484\pi$
$810$	0	0
$811$	−26782.0	−1.15961	−0.579805	−	0.814755i	$-0.696871\pi$
$811$	−26782.0	−1.15961	−0.579805	+	0.814755i	$0.696871\pi$
$812$	− 750.000i	− 0.0324136i
$813$	0	0
$814$	−19350.0	−0.833191
$815$	0	0
$816$	0	0
$817$	220.000i	0.00942084i
$818$	32259.0i	1.37886i
$819$	0	0
$820$	0	0
$821$	10110.0	0.429770	0.214885	−	0.976639i	$-0.431062\pi$
$821$	10110.0	0.429770	0.214885	+	0.976639i	$0.431062\pi$
$822$	0	0
$823$	12535.0i	0.530914i	0.964123	+	0.265457i	$0.0855229\pi$
$823$	12535.0i	0.530914i	−0.964123	+	0.265457i	$0.914477\pi$
$824$	−22260.0	−0.941097
$825$	0	0
$826$	49500.0	2.08514
$827$	9792.00i	0.411731i	0.978580	+	0.205865i	$0.0660009\pi$
$827$	9792.00i	0.411731i	−0.978580	+	0.205865i	$0.933999\pi$
$828$	0	0
$829$	4534.00	0.189955	0.0949773	−	0.995479i	$-0.469722\pi$
$829$	4534.00	0.189955	0.0949773	+	0.995479i	$0.469722\pi$
$830$	0	0
$831$	0	0
$832$	8660.00i	0.360855i
$833$	− 20304.0i	− 0.844528i
$834$	0	0
$835$	0	0
$836$	−30.0000	−0.00124111
$837$	0	0
$838$	− 8820.00i	− 0.363582i
$839$	8880.00	0.365401	0.182701	−	0.983169i	$-0.441516\pi$
$839$	8880.00	0.365401	0.182701	+	0.983169i	$0.441516\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	26088.0i	1.06776i
$843$	0	0
$844$	2086.00	0.0850747
$845$	0	0
$846$	0	0
$847$	27650.0i	1.12168i
$848$	44091.0i	1.78548i
$849$	0	0
$850$	0	0
$851$	−49020.0	−1.97460
$852$	0	0
$853$	− 2270.00i	− 0.0911176i	−0.998962	−	0.0455588i	$-0.985493\pi$
$853$	− 2270.00i	− 0.0911176i	0.998962	−	0.0455588i	$-0.0145068\pi$
$854$	−28200.0	−1.12996
$855$	0	0
$856$	−31185.0	−1.24519
$857$	− 19608.0i	− 0.781560i	−0.920484	−	0.390780i	$-0.872205\pi$
$857$	− 19608.0i	− 0.781560i	0.920484	−	0.390780i	$-0.127795\pi$
$858$	0	0
$859$	952.000	0.0378135	0.0189068	−	0.999821i	$-0.493981\pi$
$859$	952.000	0.0378135	0.0189068	+	0.999821i	$0.493981\pi$
$860$	0	0
$861$	0	0
$862$	25110.0i	0.992169i
$863$	17604.0i	0.694377i	0.937795	+	0.347188i	$0.112864\pi$
$863$	17604.0i	0.694377i	−0.937795	+	0.347188i	$0.887136\pi$
$864$	0	0
$865$	0	0
$866$	−15465.0	−0.606838
$867$	0	0
$868$	2525.00i	0.0987374i
$869$	7320.00	0.285747
$870$	0	0
$871$	−5000.00	−0.194510
$872$	18102.0i	0.702994i
$873$	0	0
$874$	−684.000	−0.0264721
$875$	0	0
$876$	0	0
$877$	21890.0i	0.842842i	0.906865	+	0.421421i	$0.138469\pi$
$877$	21890.0i	0.842842i	−0.906865	+	0.421421i	$0.861531\pi$
$878$	32961.0i	1.26695i
$879$	0	0
$880$	0	0
$881$	23940.0	0.915504	0.457752	−	0.889080i	$-0.348655\pi$
$881$	23940.0	0.915504	0.457752	+	0.889080i	$0.348655\pi$
$882$	0	0
$883$	34990.0i	1.33353i	0.745268	+	0.666765i	$0.232322\pi$
$883$	34990.0i	1.33353i	−0.745268	+	0.666765i	$0.767678\pi$
$884$	−1440.00	−0.0547878
$885$	0	0
$886$	5868.00	0.222505
$887$	− 22188.0i	− 0.839910i	−0.907545	−	0.419955i	$-0.862046\pi$
$887$	− 22188.0i	− 0.839910i	0.907545	−	0.419955i	$-0.137954\pi$
$888$	0	0
$889$	46625.0	1.75900
$890$	0	0
$891$	0	0
$892$	5240.00i	0.196691i
$893$	660.000i	0.0247324i
$894$	0	0
$895$	0	0
$896$	−41475.0	−1.54641
$897$	0	0
$898$	26190.0i	0.973242i
$899$	−3030.00	−0.112410
$900$	0	0
$901$	44712.0	1.65324
$902$	1350.00i	0.0498338i
$903$	0	0
$904$	14490.0	0.533109
$905$	0	0
$906$	0	0
$907$	37370.0i	1.36808i	0.729444	+	0.684041i	$0.239779\pi$
$907$	37370.0i	1.36808i	−0.729444	+	0.684041i	$0.760221\pi$
$908$	− 2388.00i	− 0.0872782i
$909$	0	0
$910$	0	0
$911$	40710.0	1.48055	0.740276	−	0.672303i	$-0.234695\pi$
$911$	40710.0	1.48055	0.740276	+	0.672303i	$0.234695\pi$
$912$	0	0
$913$	7335.00i	0.265885i
$914$	25995.0	0.940742
$915$	0	0
$916$	182.000	0.00656490
$917$	28875.0i	1.03984i
$918$	0	0
$919$	−20981.0	−0.753100	−0.376550	−	0.926396i	$-0.622890\pi$
$919$	−20981.0	−0.753100	−0.376550	+	0.926396i	$0.622890\pi$
$920$	0	0
$921$	0	0
$922$	− 29475.0i	− 1.05283i
$923$	7200.00i	0.256762i
$924$	0	0
$925$	0	0
$926$	−15735.0	−0.558406
$927$	0	0
$928$	1350.00i	0.0477542i
$929$	−20100.0	−0.709860	−0.354930	−	0.934893i	$-0.615495\pi$
$929$	−20100.0	−0.709860	−0.354930	+	0.934893i	$0.615495\pi$
$930$	0	0
$931$	564.000	0.0198543
$932$	450.000i	0.0158157i
$933$	0	0
$934$	33021.0	1.15683
$935$	0	0
$936$	0	0
$937$	15635.0i	0.545115i	0.962139	+	0.272558i	$0.0878696\pi$
$937$	15635.0i	0.545115i	−0.962139	+	0.272558i	$0.912130\pi$
$938$	− 18750.0i	− 0.652675i
$939$	0	0
$940$	0	0
$941$	23955.0	0.829873	0.414937	−	0.909850i	$-0.363804\pi$
$941$	23955.0	0.829873	0.414937	+	0.909850i	$0.363804\pi$
$942$	0	0
$943$	3420.00i	0.118102i
$944$	−46860.0	−1.61564
$945$	0	0
$946$	−4950.00	−0.170125
$947$	− 36393.0i	− 1.24880i	−0.781105	−	0.624400i	$-0.785344\pi$
$947$	− 36393.0i	− 1.24880i	0.781105	−	0.624400i	$-0.214656\pi$
$948$	0	0
$949$	−15700.0	−0.537032
$950$	0	0
$951$	0	0
$952$	37800.0i	1.28688i
$953$	− 43020.0i	− 1.46228i	−0.682227	−	0.731141i	$-0.738988\pi$
$953$	− 43020.0i	− 1.46228i	0.682227	−	0.731141i	$-0.261012\pi$
$954$	0	0
$955$	0	0
$956$	5190.00	0.175582
$957$	0	0
$958$	− 50850.0i	− 1.71492i
$959$	−69450.0	−2.33854
$960$	0	0
$961$	−19590.0	−0.657581
$962$	− 25800.0i	− 0.864683i
$963$	0	0
$964$	2266.00	0.0757084
$965$	0	0
$966$	0	0
$967$	− 43585.0i	− 1.44943i	−0.689049	−	0.724715i	$-0.741971\pi$
$967$	− 43585.0i	− 1.44943i	0.689049	−	0.724715i	$-0.258029\pi$
$968$	− 23226.0i	− 0.771190i
$969$	0	0
$970$	0	0
$971$	−43335.0	−1.43222	−0.716110	−	0.697987i	$-0.754079\pi$
$971$	−43335.0	−1.43222	−0.716110	+	0.697987i	$0.754079\pi$
$972$	0	0
$973$	− 48100.0i	− 1.58480i
$974$	−31920.0	−1.05008
$975$	0	0
$976$	26696.0	0.875531
$977$	30390.0i	0.995151i	0.867421	+	0.497575i	$0.165776\pi$
$977$	30390.0i	0.995151i	−0.867421	+	0.497575i	$0.834224\pi$
$978$	0	0
$979$	−6750.00	−0.220358
$980$	0	0
$981$	0	0
$982$	4905.00i	0.159394i
$983$	59226.0i	1.92168i	0.277096	+	0.960842i	$0.410628\pi$
$983$	59226.0i	1.92168i	−0.277096	+	0.960842i	$0.589372\pi$
$984$	0	0
$985$	0	0
$986$	6480.00	0.209295
$987$	0	0
$988$	− 40.0000i	− 0.00128803i
$989$	−12540.0	−0.403184
$990$	0	0
$991$	8399.00	0.269226	0.134613	−	0.990898i	$-0.457021\pi$
$991$	8399.00	0.269226	0.134613	+	0.990898i	$0.457021\pi$
$992$	− 4545.00i	− 0.145468i
$993$	0	0
$994$	−27000.0	−0.861557
$995$	0	0
$996$	0	0
$997$	13340.0i	0.423753i	0.977296	+	0.211877i	$0.0679575\pi$
$997$	13340.0i	0.423753i	−0.977296	+	0.211877i	$0.932042\pi$
$998$	47406.0i	1.50362i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.b.a.649.2		2
3.2	odd	2		675.4.b.b.649.1		2
5.2	odd	4		675.4.a.a.1.1		1
5.3	odd	4		27.4.a.b.1.1	yes	1
5.4	even	2	inner	675.4.b.a.649.1		2
15.2	even	4		675.4.a.j.1.1		1
15.8	even	4		27.4.a.a.1.1	✓	1
15.14	odd	2		675.4.b.b.649.2		2
20.3	even	4		432.4.a.n.1.1		1
35.13	even	4		1323.4.a.k.1.1		1
40.3	even	4		1728.4.a.d.1.1		1
40.13	odd	4		1728.4.a.c.1.1		1
45.13	odd	12		81.4.c.a.55.1		2
45.23	even	12		81.4.c.c.55.1		2
45.38	even	12		81.4.c.c.28.1		2
45.43	odd	12		81.4.c.a.28.1		2
60.23	odd	4		432.4.a.a.1.1		1
105.83	odd	4		1323.4.a.d.1.1		1
120.53	even	4		1728.4.a.bc.1.1		1
120.83	odd	4		1728.4.a.bd.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.a.a.1.1	✓	1	15.8	even	4
27.4.a.b.1.1	yes	1	5.3	odd	4
81.4.c.a.28.1		2	45.43	odd	12
81.4.c.a.55.1		2	45.13	odd	12
81.4.c.c.28.1		2	45.38	even	12
81.4.c.c.55.1		2	45.23	even	12
432.4.a.a.1.1		1	60.23	odd	4
432.4.a.n.1.1		1	20.3	even	4
675.4.a.a.1.1		1	5.2	odd	4
675.4.a.j.1.1		1	15.2	even	4
675.4.b.a.649.1		2	5.4	even	2	inner
675.4.b.a.649.2		2	1.1	even	1	trivial
675.4.b.b.649.1		2	3.2	odd	2
675.4.b.b.649.2		2	15.14	odd	2
1323.4.a.d.1.1		1	105.83	odd	4
1323.4.a.k.1.1		1	35.13	even	4
1728.4.a.c.1.1		1	40.13	odd	4
1728.4.a.d.1.1		1	40.3	even	4
1728.4.a.bc.1.1		1	120.53	even	4
1728.4.a.bd.1.1		1	120.83	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{2} -1.00000 q^{4} -25.0000i q^{7} +21.0000i q^{8} +O(q^{10})\)	\(q+3.00000i q^{2} -1.00000 q^{4} -25.0000i q^{7} +21.0000i q^{8} -15.0000 q^{11} -20.0000i q^{13} +75.0000 q^{14} -71.0000 q^{16} +72.0000i q^{17} -2.00000 q^{19} -45.0000i q^{22} -114.000i q^{23} +60.0000 q^{26} +25.0000i q^{28} -30.0000 q^{29} +101.000 q^{31} -45.0000i q^{32} -216.000 q^{34} -430.000i q^{37} -6.00000i q^{38} -30.0000 q^{41} -110.000i q^{43} +15.0000 q^{44} +342.000 q^{46} -330.000i q^{47} -282.000 q^{49} +20.0000i q^{52} -621.000i q^{53} +525.000 q^{56} -90.0000i q^{58} +660.000 q^{59} -376.000 q^{61} +303.000i q^{62} -433.000 q^{64} -250.000i q^{67} -72.0000i q^{68} -360.000 q^{71} -785.000i q^{73} +1290.00 q^{74} +2.00000 q^{76} +375.000i q^{77} -488.000 q^{79} -90.0000i q^{82} -489.000i q^{83} +330.000 q^{86} -315.000i q^{88} +450.000 q^{89} -500.000 q^{91} +114.000i q^{92} +990.000 q^{94} -1105.00i q^{97} -846.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 30 q^{11} + 150 q^{14} - 142 q^{16} - 4 q^{19} + 120 q^{26} - 60 q^{29} + 202 q^{31} - 432 q^{34} - 60 q^{41} + 30 q^{44} + 684 q^{46} - 564 q^{49} + 1050 q^{56} + 1320 q^{59} - 752 q^{61} - 866 q^{64} - 720 q^{71} + 2580 q^{74} + 4 q^{76} - 976 q^{79} + 660 q^{86} + 900 q^{89} - 1000 q^{91} + 1980 q^{94}+O(q^{100}) \) 2 * q - 2 * q^4 - 30 * q^11 + 150 * q^14 - 142 * q^16 - 4 * q^19 + 120 * q^26 - 60 * q^29 + 202 * q^31 - 432 * q^34 - 60 * q^41 + 30 * q^44 + 684 * q^46 - 564 * q^49 + 1050 * q^56 + 1320 * q^59 - 752 * q^61 - 866 * q^64 - 720 * q^71 + 2580 * q^74 + 4 * q^76 - 976 * q^79 + 660 * q^86 + 900 * q^89 - 1000 * q^91 + 1980 * q^94

Introduction

L-functions

Modular forms

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Representations

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Database

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