LMFDB - Embedded newform 1350.4.c.e.649.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(649,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("1350.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1350.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:	$79.6525785077$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1350.649
Dual form			1350.4.c.e.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000i q^{2} -4.00000 q^{4} +23.0000i q^{7} -8.00000i q^{8} +O(q^{10})$	$q+2.00000i q^{2} -4.00000 q^{4} +23.0000i q^{7} -8.00000i q^{8} -30.0000 q^{11} +34.0000i q^{13} -46.0000 q^{14} +16.0000 q^{16} +42.0000i q^{17} +139.000 q^{19} -60.0000i q^{22} +192.000i q^{23} -68.0000 q^{26} -92.0000i q^{28} +234.000 q^{29} -55.0000 q^{31} +32.0000i q^{32} -84.0000 q^{34} +191.000i q^{37} +278.000i q^{38} -138.000 q^{41} -53.0000i q^{43} +120.000 q^{44} -384.000 q^{46} -366.000i q^{47} -186.000 q^{49} -136.000i q^{52} -330.000i q^{53} +184.000 q^{56} +468.000i q^{58} -396.000 q^{59} +23.0000 q^{61} -110.000i q^{62} -64.0000 q^{64} +452.000i q^{67} -168.000i q^{68} -204.000 q^{71} +691.000i q^{73} -382.000 q^{74} -556.000 q^{76} -690.000i q^{77} +709.000 q^{79} -276.000i q^{82} +1098.00i q^{83} +106.000 q^{86} +240.000i q^{88} -816.000 q^{89} -782.000 q^{91} -768.000i q^{92} +732.000 q^{94} +905.000i q^{97} -372.000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{4}+O(q^{10}) $ 2 * q - 8 * q^4	$ 2 q - 8 q^{4} - 60 q^{11} - 92 q^{14} + 32 q^{16} + 278 q^{19} - 136 q^{26} + 468 q^{29} - 110 q^{31} - 168 q^{34} - 276 q^{41} + 240 q^{44} - 768 q^{46} - 372 q^{49} + 368 q^{56} - 792 q^{59} + 46 q^{61} - 128 q^{64} - 408 q^{71} - 764 q^{74} - 1112 q^{76} + 1418 q^{79} + 212 q^{86} - 1632 q^{89} - 1564 q^{91} + 1464 q^{94}+O(q^{100}) $ 2 * q - 8 * q^4 - 60 * q^11 - 92 * q^14 + 32 * q^16 + 278 * q^19 - 136 * q^26 + 468 * q^29 - 110 * q^31 - 168 * q^34 - 276 * q^41 + 240 * q^44 - 768 * q^46 - 372 * q^49 + 368 * q^56 - 792 * q^59 + 46 * q^61 - 128 * q^64 - 408 * q^71 - 764 * q^74 - 1112 * q^76 + 1418 * q^79 + 212 * q^86 - 1632 * q^89 - 1564 * q^91 + 1464 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.00000i	0.707107i
$2$	2.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	23.0000i	1.24188i	0.783857	+	0.620942i	$0.213250\pi$
$7$	23.0000i	1.24188i	−0.783857	+	0.620942i	$0.786750\pi$
$8$	− 8.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	−30.0000	−0.822304	−0.411152	−	0.911567i	$-0.634873\pi$
$11$	−30.0000	−0.822304	−0.411152	+	0.911567i	$0.634873\pi$
$12$	0	0
$13$	34.0000i	0.725377i	0.931910	+	0.362689i	$0.118141\pi$
$13$	34.0000i	0.725377i	−0.931910	+	0.362689i	$0.881859\pi$
$14$	−46.0000	−0.878144
$15$	0	0
$16$	16.0000	0.250000
$17$	42.0000i	0.599206i	0.954064	+	0.299603i	$0.0968542\pi$
$17$	42.0000i	0.599206i	−0.954064	+	0.299603i	$0.903146\pi$
$18$	0	0
$19$	139.000	1.67836	0.839179	−	0.543856i	$-0.183036\pi$
$19$	139.000	1.67836	0.839179	+	0.543856i	$0.183036\pi$
$20$	0	0
$21$	0	0
$22$	− 60.0000i	− 0.581456i
$23$	192.000i	1.74064i	0.492485	+	0.870321i	$0.336089\pi$
$23$	192.000i	1.74064i	−0.492485	+	0.870321i	$0.663911\pi$
$24$	0	0
$25$	0	0
$26$	−68.0000	−0.512919
$27$	0	0
$28$	− 92.0000i	− 0.620942i
$29$	234.000	1.49837	0.749185	−	0.662361i	$-0.230446\pi$
$29$	234.000	1.49837	0.749185	+	0.662361i	$0.230446\pi$
$30$	0	0
$31$	−55.0000	−0.318655	−0.159327	−	0.987226i	$-0.550933\pi$
$31$	−55.0000	−0.318655	−0.159327	+	0.987226i	$0.550933\pi$
$32$	32.0000i	0.176777i
$33$	0	0
$34$	−84.0000	−0.423702
$35$	0	0
$36$	0	0
$37$	191.000i	0.848654i	0.905509	+	0.424327i	$0.139489\pi$
$37$	191.000i	0.848654i	−0.905509	+	0.424327i	$0.860511\pi$
$38$	278.000i	1.18678i
$39$	0	0
$40$	0	0
$41$	−138.000	−0.525658	−0.262829	−	0.964842i	$-0.584656\pi$
$41$	−138.000	−0.525658	−0.262829	+	0.964842i	$0.584656\pi$
$42$	0	0
$43$	− 53.0000i	− 0.187963i	−0.995574	−	0.0939817i	$-0.970040\pi$
$43$	− 53.0000i	− 0.187963i	0.995574	−	0.0939817i	$-0.0299595\pi$
$44$	120.000	0.411152
$45$	0	0
$46$	−384.000	−1.23082
$47$	− 366.000i	− 1.13588i	−0.823068	−	0.567942i	$-0.807740\pi$
$47$	− 366.000i	− 1.13588i	0.823068	−	0.567942i	$-0.192260\pi$
$48$	0	0
$49$	−186.000	−0.542274
$50$	0	0
$51$	0	0
$52$	− 136.000i	− 0.362689i
$53$	− 330.000i	− 0.855264i	−0.903953	−	0.427632i	$-0.859348\pi$
$53$	− 330.000i	− 0.855264i	0.903953	−	0.427632i	$-0.140652\pi$
$54$	0	0
$55$	0	0
$56$	184.000	0.439072
$57$	0	0
$58$	468.000i	1.05951i
$59$	−396.000	−0.873810	−0.436905	−	0.899508i	$-0.643925\pi$
$59$	−396.000	−0.873810	−0.436905	+	0.899508i	$0.643925\pi$
$60$	0	0
$61$	23.0000	0.0482762	0.0241381	−	0.999709i	$-0.492316\pi$
$61$	23.0000	0.0482762	0.0241381	+	0.999709i	$0.492316\pi$
$62$	− 110.000i	− 0.225323i
$63$	0	0
$64$	−64.0000	−0.125000
$65$	0	0
$66$	0	0
$67$	452.000i	0.824188i	0.911141	+	0.412094i	$0.135202\pi$
$67$	452.000i	0.824188i	−0.911141	+	0.412094i	$0.864798\pi$
$68$	− 168.000i	− 0.299603i
$69$	0	0
$70$	0	0
$71$	−204.000	−0.340991	−0.170495	−	0.985358i	$-0.554537\pi$
$71$	−204.000	−0.340991	−0.170495	+	0.985358i	$0.554537\pi$
$72$	0	0
$73$	691.000i	1.10788i	0.832556	+	0.553941i	$0.186877\pi$
$73$	691.000i	1.10788i	−0.832556	+	0.553941i	$0.813123\pi$
$74$	−382.000	−0.600089
$75$	0	0
$76$	−556.000	−0.839179
$77$	− 690.000i	− 1.02121i
$78$	0	0
$79$	709.000	1.00973	0.504865	−	0.863198i	$-0.331542\pi$
$79$	709.000	1.00973	0.504865	+	0.863198i	$0.331542\pi$
$80$	0	0
$81$	0	0
$82$	− 276.000i	− 0.371696i
$83$	1098.00i	1.45206i	0.687662	+	0.726031i	$0.258637\pi$
$83$	1098.00i	1.45206i	−0.687662	+	0.726031i	$0.741363\pi$
$84$	0	0
$85$	0	0
$86$	106.000	0.132910
$87$	0	0
$88$	240.000i	0.290728i
$89$	−816.000	−0.971863	−0.485932	−	0.873997i	$-0.661520\pi$
$89$	−816.000	−0.971863	−0.485932	+	0.873997i	$0.661520\pi$
$90$	0	0
$91$	−782.000	−0.900834
$92$	− 768.000i	− 0.870321i
$93$	0	0
$94$	732.000	0.803192
$95$	0	0
$96$	0	0
$97$	905.000i	0.947308i	0.880711	+	0.473654i	$0.157065\pi$
$97$	905.000i	0.947308i	−0.880711	+	0.473654i	$0.842935\pi$
$98$	− 372.000i	− 0.383446i
$99$	0	0
$100$	0	0
$101$	1278.00	1.25907	0.629533	−	0.776973i	$-0.283246\pi$
$101$	1278.00	1.25907	0.629533	+	0.776973i	$0.283246\pi$
$102$	0	0
$103$	− 605.000i	− 0.578761i	−0.957214	−	0.289381i	$-0.906551\pi$
$103$	− 605.000i	− 0.578761i	0.957214	−	0.289381i	$-0.0934493\pi$
$104$	272.000	0.256460
$105$	0	0
$106$	660.000	0.604763
$107$	− 1488.00i	− 1.34440i	−0.740371	−	0.672198i	$-0.765350\pi$
$107$	− 1488.00i	− 1.34440i	0.740371	−	0.672198i	$-0.234650\pi$
$108$	0	0
$109$	−593.000	−0.521093	−0.260546	−	0.965461i	$-0.583903\pi$
$109$	−593.000	−0.521093	−0.260546	+	0.965461i	$0.583903\pi$
$110$	0	0
$111$	0	0
$112$	368.000i	0.310471i
$113$	324.000i	0.269729i	0.990864	+	0.134864i	$0.0430599\pi$
$113$	324.000i	0.269729i	−0.990864	+	0.134864i	$0.956940\pi$
$114$	0	0
$115$	0	0
$116$	−936.000	−0.749185
$117$	0	0
$118$	− 792.000i	− 0.617877i
$119$	−966.000	−0.744143
$120$	0	0
$121$	−431.000	−0.323817
$122$	46.0000i	0.0341364i
$123$	0	0
$124$	220.000	0.159327
$125$	0	0
$126$	0	0
$127$	1928.00i	1.34711i	0.739139	+	0.673553i	$0.235232\pi$
$127$	1928.00i	1.34711i	−0.739139	+	0.673553i	$0.764768\pi$
$128$	− 128.000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	−2742.00	−1.82878	−0.914388	−	0.404839i	$-0.867328\pi$
$131$	−2742.00	−1.82878	−0.914388	+	0.404839i	$0.867328\pi$
$132$	0	0
$133$	3197.00i	2.08432i
$134$	−904.000	−0.582789
$135$	0	0
$136$	336.000	0.211851
$137$	− 1326.00i	− 0.826918i	−0.910523	−	0.413459i	$-0.864320\pi$
$137$	− 1326.00i	− 0.826918i	0.910523	−	0.413459i	$-0.135680\pi$
$138$	0	0
$139$	−893.000	−0.544916	−0.272458	−	0.962168i	$-0.587837\pi$
$139$	−893.000	−0.544916	−0.272458	+	0.962168i	$0.587837\pi$
$140$	0	0
$141$	0	0
$142$	− 408.000i	− 0.241117i
$143$	− 1020.00i	− 0.596480i
$144$	0	0
$145$	0	0
$146$	−1382.00	−0.783391
$147$	0	0
$148$	− 764.000i	− 0.424327i
$149$	−2502.00	−1.37565	−0.687825	−	0.725877i	$-0.741434\pi$
$149$	−2502.00	−1.37565	−0.687825	+	0.725877i	$0.741434\pi$
$150$	0	0
$151$	−2767.00	−1.49123	−0.745613	−	0.666379i	$-0.767843\pi$
$151$	−2767.00	−1.49123	−0.745613	+	0.666379i	$0.767843\pi$
$152$	− 1112.00i	− 0.593389i
$153$	0	0
$154$	1380.00	0.722101
$155$	0	0
$156$	0	0
$157$	− 2701.00i	− 1.37301i	−0.727123	−	0.686507i	$-0.759143\pi$
$157$	− 2701.00i	− 1.37301i	0.727123	−	0.686507i	$-0.240857\pi$
$158$	1418.00i	0.713987i
$159$	0	0
$160$	0	0
$161$	−4416.00	−2.16167
$162$	0	0
$163$	− 1748.00i	− 0.839963i	−0.907533	−	0.419981i	$-0.862037\pi$
$163$	− 1748.00i	− 0.839963i	0.907533	−	0.419981i	$-0.137963\pi$
$164$	552.000	0.262829
$165$	0	0
$166$	−2196.00	−1.02676
$167$	534.000i	0.247438i	0.992317	+	0.123719i	$0.0394822\pi$
$167$	534.000i	0.247438i	−0.992317	+	0.123719i	$0.960518\pi$
$168$	0	0
$169$	1041.00	0.473828
$170$	0	0
$171$	0	0
$172$	212.000i	0.0939817i
$173$	− 192.000i	− 0.0843786i	−0.999110	−	0.0421893i	$-0.986567\pi$
$173$	− 192.000i	− 0.0843786i	0.999110	−	0.0421893i	$-0.0134333\pi$
$174$	0	0
$175$	0	0
$176$	−480.000	−0.205576
$177$	0	0
$178$	− 1632.00i	− 0.687211i
$179$	1140.00	0.476020	0.238010	−	0.971263i	$-0.423505\pi$
$179$	1140.00	0.476020	0.238010	+	0.971263i	$0.423505\pi$
$180$	0	0
$181$	398.000	0.163443	0.0817213	−	0.996655i	$-0.473958\pi$
$181$	398.000	0.163443	0.0817213	+	0.996655i	$0.473958\pi$
$182$	− 1564.00i	− 0.636986i
$183$	0	0
$184$	1536.00	0.615410
$185$	0	0
$186$	0	0
$187$	− 1260.00i	− 0.492729i
$188$	1464.00i	0.567942i
$189$	0	0
$190$	0	0
$191$	−3474.00	−1.31607	−0.658036	−	0.752986i	$-0.728613\pi$
$191$	−3474.00	−1.31607	−0.658036	+	0.752986i	$0.728613\pi$
$192$	0	0
$193$	2713.00i	1.01184i	0.862579	+	0.505922i	$0.168848\pi$
$193$	2713.00i	1.01184i	−0.862579	+	0.505922i	$0.831152\pi$
$194$	−1810.00	−0.669848
$195$	0	0
$196$	744.000	0.271137
$197$	− 4734.00i	− 1.71210i	−0.516894	−	0.856050i	$-0.672912\pi$
$197$	− 4734.00i	− 1.71210i	0.516894	−	0.856050i	$-0.327088\pi$
$198$	0	0
$199$	−5132.00	−1.82813	−0.914065	−	0.405568i	$-0.867074\pi$
$199$	−5132.00	−1.82813	−0.914065	+	0.405568i	$0.867074\pi$
$200$	0	0
$201$	0	0
$202$	2556.00i	0.890295i
$203$	5382.00i	1.86080i
$204$	0	0
$205$	0	0
$206$	1210.00	0.409246
$207$	0	0
$208$	544.000i	0.181344i
$209$	−4170.00	−1.38012
$210$	0	0
$211$	5240.00	1.70965	0.854826	−	0.518915i	$-0.173664\pi$
$211$	5240.00	1.70965	0.854826	+	0.518915i	$0.173664\pi$
$212$	1320.00i	0.427632i
$213$	0	0
$214$	2976.00	0.950632
$215$	0	0
$216$	0	0
$217$	− 1265.00i	− 0.395732i
$218$	− 1186.00i	− 0.368468i
$219$	0	0
$220$	0	0
$221$	−1428.00	−0.434650
$222$	0	0
$223$	4519.00i	1.35702i	0.734593	+	0.678508i	$0.237373\pi$
$223$	4519.00i	1.35702i	−0.734593	+	0.678508i	$0.762627\pi$
$224$	−736.000	−0.219536
$225$	0	0
$226$	−648.000	−0.190727
$227$	− 5064.00i	− 1.48066i	−0.672244	−	0.740329i	$-0.734670\pi$
$227$	− 5064.00i	− 1.48066i	0.672244	−	0.740329i	$-0.265330\pi$
$228$	0	0
$229$	−2573.00	−0.742483	−0.371242	−	0.928536i	$-0.621068\pi$
$229$	−2573.00	−0.742483	−0.371242	+	0.928536i	$0.621068\pi$
$230$	0	0
$231$	0	0
$232$	− 1872.00i	− 0.529754i
$233$	4098.00i	1.15223i	0.817370	+	0.576114i	$0.195431\pi$
$233$	4098.00i	1.15223i	−0.817370	+	0.576114i	$0.804569\pi$
$234$	0	0
$235$	0	0
$236$	1584.00	0.436905
$237$	0	0
$238$	− 1932.00i	− 0.526189i
$239$	−306.000	−0.0828180	−0.0414090	−	0.999142i	$-0.513185\pi$
$239$	−306.000	−0.0828180	−0.0414090	+	0.999142i	$0.513185\pi$
$240$	0	0
$241$	6482.00	1.73254	0.866270	−	0.499575i	$-0.166511\pi$
$241$	6482.00	1.73254	0.866270	+	0.499575i	$0.166511\pi$
$242$	− 862.000i	− 0.228973i
$243$	0	0
$244$	−92.0000	−0.0241381
$245$	0	0
$246$	0	0
$247$	4726.00i	1.21744i
$248$	440.000i	0.112661i
$249$	0	0
$250$	0	0
$251$	5850.00	1.47111	0.735555	−	0.677465i	$-0.236921\pi$
$251$	5850.00	1.47111	0.735555	+	0.677465i	$0.236921\pi$
$252$	0	0
$253$	− 5760.00i	− 1.43134i
$254$	−3856.00	−0.952547
$255$	0	0
$256$	256.000	0.0625000
$257$	− 5598.00i	− 1.35873i	−0.733800	−	0.679365i	$-0.762255\pi$
$257$	− 5598.00i	− 1.35873i	0.733800	−	0.679365i	$-0.237745\pi$
$258$	0	0
$259$	−4393.00	−1.05393
$260$	0	0
$261$	0	0
$262$	− 5484.00i	− 1.29314i
$263$	− 8286.00i	− 1.94272i	−0.237603	−	0.971362i	$-0.576362\pi$
$263$	− 8286.00i	− 1.94272i	0.237603	−	0.971362i	$-0.423638\pi$
$264$	0	0
$265$	0	0
$266$	−6394.00	−1.47384
$267$	0	0
$268$	− 1808.00i	− 0.412094i
$269$	−504.000	−0.114236	−0.0571179	−	0.998367i	$-0.518191\pi$
$269$	−504.000	−0.114236	−0.0571179	+	0.998367i	$0.518191\pi$
$270$	0	0
$271$	−4489.00	−1.00623	−0.503113	−	0.864221i	$-0.667812\pi$
$271$	−4489.00	−1.00623	−0.503113	+	0.864221i	$0.667812\pi$
$272$	672.000i	0.149801i
$273$	0	0
$274$	2652.00	0.584720
$275$	0	0
$276$	0	0
$277$	2213.00i	0.480023i	0.970770	+	0.240011i	$0.0771512\pi$
$277$	2213.00i	0.480023i	−0.970770	+	0.240011i	$0.922849\pi$
$278$	− 1786.00i	− 0.385314i
$279$	0	0
$280$	0	0
$281$	2718.00	0.577019	0.288509	−	0.957477i	$-0.406840\pi$
$281$	2718.00	0.577019	0.288509	+	0.957477i	$0.406840\pi$
$282$	0	0
$283$	− 5615.00i	− 1.17942i	−0.807613	−	0.589712i	$-0.799241\pi$
$283$	− 5615.00i	− 1.17942i	0.807613	−	0.589712i	$-0.200759\pi$
$284$	816.000	0.170495
$285$	0	0
$286$	2040.00	0.421775
$287$	− 3174.00i	− 0.652806i
$288$	0	0
$289$	3149.00	0.640953
$290$	0	0
$291$	0	0
$292$	− 2764.00i	− 0.553941i
$293$	1488.00i	0.296689i	0.988936	+	0.148345i	$0.0473945\pi$
$293$	1488.00i	0.296689i	−0.988936	+	0.148345i	$0.952606\pi$
$294$	0	0
$295$	0	0
$296$	1528.00	0.300045
$297$	0	0
$298$	− 5004.00i	− 0.972731i
$299$	−6528.00	−1.26262
$300$	0	0
$301$	1219.00	0.233429
$302$	− 5534.00i	− 1.05446i
$303$	0	0
$304$	2224.00	0.419589
$305$	0	0
$306$	0	0
$307$	7487.00i	1.39188i	0.718102	+	0.695938i	$0.245011\pi$
$307$	7487.00i	1.39188i	−0.718102	+	0.695938i	$0.754989\pi$
$308$	2760.00i	0.510603i
$309$	0	0
$310$	0	0
$311$	−8118.00	−1.48016	−0.740080	−	0.672519i	$-0.765212\pi$
$311$	−8118.00	−1.48016	−0.740080	+	0.672519i	$0.765212\pi$
$312$	0	0
$313$	5002.00i	0.903290i	0.892198	+	0.451645i	$0.149163\pi$
$313$	5002.00i	0.903290i	−0.892198	+	0.451645i	$0.850837\pi$
$314$	5402.00	0.970868
$315$	0	0
$316$	−2836.00	−0.504865
$317$	− 5082.00i	− 0.900421i	−0.892922	−	0.450211i	$-0.851349\pi$
$317$	− 5082.00i	− 0.900421i	0.892922	−	0.450211i	$-0.148651\pi$
$318$	0	0
$319$	−7020.00	−1.23211
$320$	0	0
$321$	0	0
$322$	− 8832.00i	− 1.52853i
$323$	5838.00i	1.00568i
$324$	0	0
$325$	0	0
$326$	3496.00	0.593943
$327$	0	0
$328$	1104.00i	0.185848i
$329$	8418.00	1.41064
$330$	0	0
$331$	7625.00	1.26619	0.633094	−	0.774075i	$-0.281785\pi$
$331$	7625.00	1.26619	0.633094	+	0.774075i	$0.281785\pi$
$332$	− 4392.00i	− 0.726031i
$333$	0	0
$334$	−1068.00	−0.174965
$335$	0	0
$336$	0	0
$337$	− 3778.00i	− 0.610685i	−0.952243	−	0.305342i	$-0.901229\pi$
$337$	− 3778.00i	− 0.610685i	0.952243	−	0.305342i	$-0.0987709\pi$
$338$	2082.00i	0.335047i
$339$	0	0
$340$	0	0
$341$	1650.00	0.262031
$342$	0	0
$343$	3611.00i	0.568442i
$344$	−424.000	−0.0664551
$345$	0	0
$346$	384.000	0.0596646
$347$	8268.00i	1.27911i	0.768747	+	0.639553i	$0.220880\pi$
$347$	8268.00i	1.27911i	−0.768747	+	0.639553i	$0.779120\pi$
$348$	0	0
$349$	−1379.00	−0.211508	−0.105754	−	0.994392i	$-0.533726\pi$
$349$	−1379.00	−0.211508	−0.105754	+	0.994392i	$0.533726\pi$
$350$	0	0
$351$	0	0
$352$	− 960.000i	− 0.145364i
$353$	3072.00i	0.463190i	0.972812	+	0.231595i	$0.0743944\pi$
$353$	3072.00i	0.463190i	−0.972812	+	0.231595i	$0.925606\pi$
$354$	0	0
$355$	0	0
$356$	3264.00	0.485932
$357$	0	0
$358$	2280.00i	0.336597i
$359$	−7446.00	−1.09467	−0.547333	−	0.836915i	$-0.684357\pi$
$359$	−7446.00	−1.09467	−0.547333	+	0.836915i	$0.684357\pi$
$360$	0	0
$361$	12462.0	1.81688
$362$	796.000i	0.115571i
$363$	0	0
$364$	3128.00	0.450417
$365$	0	0
$366$	0	0
$367$	− 6496.00i	− 0.923947i	−0.886894	−	0.461973i	$-0.847142\pi$
$367$	− 6496.00i	− 0.923947i	0.886894	−	0.461973i	$-0.152858\pi$
$368$	3072.00i	0.435161i
$369$	0	0
$370$	0	0
$371$	7590.00	1.06214
$372$	0	0
$373$	1633.00i	0.226685i	0.993556	+	0.113343i	$0.0361557\pi$
$373$	1633.00i	0.226685i	−0.993556	+	0.113343i	$0.963844\pi$
$374$	2520.00	0.348412
$375$	0	0
$376$	−2928.00	−0.401596
$377$	7956.00i	1.08688i
$378$	0	0
$379$	−6788.00	−0.919990	−0.459995	−	0.887922i	$-0.652149\pi$
$379$	−6788.00	−0.919990	−0.459995	+	0.887922i	$0.652149\pi$
$380$	0	0
$381$	0	0
$382$	− 6948.00i	− 0.930604i
$383$	6582.00i	0.878132i	0.898455	+	0.439066i	$0.144691\pi$
$383$	6582.00i	0.878132i	−0.898455	+	0.439066i	$0.855309\pi$
$384$	0	0
$385$	0	0
$386$	−5426.00	−0.715482
$387$	0	0
$388$	− 3620.00i	− 0.473654i
$389$	2850.00	0.371467	0.185734	−	0.982600i	$-0.440534\pi$
$389$	2850.00	0.371467	0.185734	+	0.982600i	$0.440534\pi$
$390$	0	0
$391$	−8064.00	−1.04300
$392$	1488.00i	0.191723i
$393$	0	0
$394$	9468.00	1.21064
$395$	0	0
$396$	0	0
$397$	7451.00i	0.941952i	0.882146	+	0.470976i	$0.156098\pi$
$397$	7451.00i	0.941952i	−0.882146	+	0.470976i	$0.843902\pi$
$398$	− 10264.0i	− 1.29268i
$399$	0	0
$400$	0	0
$401$	−14124.0	−1.75890	−0.879450	−	0.475991i	$-0.842089\pi$
$401$	−14124.0	−1.75890	−0.879450	+	0.475991i	$0.842089\pi$
$402$	0	0
$403$	− 1870.00i	− 0.231145i
$404$	−5112.00	−0.629533
$405$	0	0
$406$	−10764.0	−1.31578
$407$	− 5730.00i	− 0.697851i
$408$	0	0
$409$	−6374.00	−0.770597	−0.385298	−	0.922792i	$-0.625901\pi$
$409$	−6374.00	−0.770597	−0.385298	+	0.922792i	$0.625901\pi$
$410$	0	0
$411$	0	0
$412$	2420.00i	0.289381i
$413$	− 9108.00i	− 1.08517i
$414$	0	0
$415$	0	0
$416$	−1088.00	−0.128230
$417$	0	0
$418$	− 8340.00i	− 0.975892i
$419$	3948.00	0.460316	0.230158	−	0.973153i	$-0.426076\pi$
$419$	3948.00	0.460316	0.230158	+	0.973153i	$0.426076\pi$
$420$	0	0
$421$	12629.0	1.46199	0.730997	−	0.682380i	$-0.239055\pi$
$421$	12629.0	1.46199	0.730997	+	0.682380i	$0.239055\pi$
$422$	10480.0i	1.20891i
$423$	0	0
$424$	−2640.00	−0.302381
$425$	0	0
$426$	0	0
$427$	529.000i	0.0599534i
$428$	5952.00i	0.672198i
$429$	0	0
$430$	0	0
$431$	−4842.00	−0.541139	−0.270570	−	0.962700i	$-0.587212\pi$
$431$	−4842.00	−0.541139	−0.270570	+	0.962700i	$0.587212\pi$
$432$	0	0
$433$	− 3851.00i	− 0.427407i	−0.976899	−	0.213704i	$-0.931447\pi$
$433$	− 3851.00i	− 0.427407i	0.976899	−	0.213704i	$-0.0685526\pi$
$434$	2530.00	0.279825
$435$	0	0
$436$	2372.00	0.260546
$437$	26688.0i	2.92142i
$438$	0	0
$439$	7435.00	0.808322	0.404161	−	0.914688i	$-0.367564\pi$
$439$	7435.00	0.808322	0.404161	+	0.914688i	$0.367564\pi$
$440$	0	0
$441$	0	0
$442$	− 2856.00i	− 0.307344i
$443$	− 5760.00i	− 0.617756i	−0.951102	−	0.308878i	$-0.900047\pi$
$443$	− 5760.00i	− 0.617756i	0.951102	−	0.308878i	$-0.0999535\pi$
$444$	0	0
$445$	0	0
$446$	−9038.00	−0.959555
$447$	0	0
$448$	− 1472.00i	− 0.155235i
$449$	−2190.00	−0.230184	−0.115092	−	0.993355i	$-0.536716\pi$
$449$	−2190.00	−0.230184	−0.115092	+	0.993355i	$0.536716\pi$
$450$	0	0
$451$	4140.00	0.432251
$452$	− 1296.00i	− 0.134864i
$453$	0	0
$454$	10128.0	1.04698
$455$	0	0
$456$	0	0
$457$	7202.00i	0.737189i	0.929590	+	0.368594i	$0.120161\pi$
$457$	7202.00i	0.737189i	−0.929590	+	0.368594i	$0.879839\pi$
$458$	− 5146.00i	− 0.525015i
$459$	0	0
$460$	0	0
$461$	13476.0	1.36147	0.680737	−	0.732528i	$-0.261659\pi$
$461$	13476.0	1.36147	0.680737	+	0.732528i	$0.261659\pi$
$462$	0	0
$463$	10843.0i	1.08837i	0.838964	+	0.544187i	$0.183162\pi$
$463$	10843.0i	1.08837i	−0.838964	+	0.544187i	$0.816838\pi$
$464$	3744.00	0.374592
$465$	0	0
$466$	−8196.00	−0.814748
$467$	− 6108.00i	− 0.605235i	−0.953112	−	0.302617i	$-0.902140\pi$
$467$	− 6108.00i	− 0.605235i	0.953112	−	0.302617i	$-0.0978604\pi$
$468$	0	0
$469$	−10396.0	−1.02355
$470$	0	0
$471$	0	0
$472$	3168.00i	0.308939i
$473$	1590.00i	0.154563i
$474$	0	0
$475$	0	0
$476$	3864.00	0.372072
$477$	0	0
$478$	− 612.000i	− 0.0585611i
$479$	9852.00	0.939769	0.469885	−	0.882728i	$-0.344296\pi$
$479$	9852.00	0.939769	0.469885	+	0.882728i	$0.344296\pi$
$480$	0	0
$481$	−6494.00	−0.615594
$482$	12964.0i	1.22509i
$483$	0	0
$484$	1724.00	0.161908
$485$	0	0
$486$	0	0
$487$	7796.00i	0.725401i	0.931906	+	0.362701i	$0.118145\pi$
$487$	7796.00i	0.725401i	−0.931906	+	0.362701i	$0.881855\pi$
$488$	− 184.000i	− 0.0170682i
$489$	0	0
$490$	0	0
$491$	−2454.00	−0.225555	−0.112777	−	0.993620i	$-0.535975\pi$
$491$	−2454.00	−0.225555	−0.112777	+	0.993620i	$0.535975\pi$
$492$	0	0
$493$	9828.00i	0.897831i
$494$	−9452.00	−0.860862
$495$	0	0
$496$	−880.000	−0.0796636
$497$	− 4692.00i	− 0.423471i
$498$	0	0
$499$	2953.00	0.264919	0.132459	−	0.991188i	$-0.457713\pi$
$499$	2953.00	0.264919	0.132459	+	0.991188i	$0.457713\pi$
$500$	0	0
$501$	0	0
$502$	11700.0i	1.04023i
$503$	− 11322.0i	− 1.00362i	−0.864977	−	0.501812i	$-0.832667\pi$
$503$	− 11322.0i	− 1.00362i	0.864977	−	0.501812i	$-0.167333\pi$
$504$	0	0
$505$	0	0
$506$	11520.0	1.01211
$507$	0	0
$508$	− 7712.00i	− 0.673553i
$509$	9696.00	0.844337	0.422169	−	0.906517i	$-0.361269\pi$
$509$	9696.00	0.844337	0.422169	+	0.906517i	$0.361269\pi$
$510$	0	0
$511$	−15893.0	−1.37586
$512$	512.000i	0.0441942i
$513$	0	0
$514$	11196.0	0.960767
$515$	0	0
$516$	0	0
$517$	10980.0i	0.934042i
$518$	− 8786.00i	− 0.745241i
$519$	0	0
$520$	0	0
$521$	12192.0	1.02522	0.512612	−	0.858621i	$-0.328678\pi$
$521$	12192.0	1.02522	0.512612	+	0.858621i	$0.328678\pi$
$522$	0	0
$523$	8491.00i	0.709915i	0.934882	+	0.354957i	$0.115505\pi$
$523$	8491.00i	0.709915i	−0.934882	+	0.354957i	$0.884495\pi$
$524$	10968.0	0.914388
$525$	0	0
$526$	16572.0	1.37371
$527$	− 2310.00i	− 0.190940i
$528$	0	0
$529$	−24697.0	−2.02983
$530$	0	0
$531$	0	0
$532$	− 12788.0i	− 1.04216i
$533$	− 4692.00i	− 0.381300i
$534$	0	0
$535$	0	0
$536$	3616.00	0.291394
$537$	0	0
$538$	− 1008.00i	− 0.0807769i
$539$	5580.00	0.445914
$540$	0	0
$541$	−9355.00	−0.743443	−0.371722	−	0.928344i	$-0.621232\pi$
$541$	−9355.00	−0.743443	−0.371722	+	0.928344i	$0.621232\pi$
$542$	− 8978.00i	− 0.711509i
$543$	0	0
$544$	−1344.00	−0.105926
$545$	0	0
$546$	0	0
$547$	− 295.000i	− 0.0230590i	−0.999934	−	0.0115295i	$-0.996330\pi$
$547$	− 295.000i	− 0.0230590i	0.999934	−	0.0115295i	$-0.00367004\pi$
$548$	5304.00i	0.413459i
$549$	0	0
$550$	0	0
$551$	32526.0	2.51480
$552$	0	0
$553$	16307.0i	1.25397i
$554$	−4426.00	−0.339427
$555$	0	0
$556$	3572.00	0.272458
$557$	16914.0i	1.28666i	0.765589	+	0.643330i	$0.222448\pi$
$557$	16914.0i	1.28666i	−0.765589	+	0.643330i	$0.777552\pi$
$558$	0	0
$559$	1802.00	0.136344
$560$	0	0
$561$	0	0
$562$	5436.00i	0.408014i
$563$	− 12108.0i	− 0.906379i	−0.891414	−	0.453189i	$-0.850286\pi$
$563$	− 12108.0i	− 0.906379i	0.891414	−	0.453189i	$-0.149714\pi$
$564$	0	0
$565$	0	0
$566$	11230.0	0.833979
$567$	0	0
$568$	1632.00i	0.120558i
$569$	−6960.00	−0.512792	−0.256396	−	0.966572i	$-0.582535\pi$
$569$	−6960.00	−0.512792	−0.256396	+	0.966572i	$0.582535\pi$
$570$	0	0
$571$	−10687.0	−0.783252	−0.391626	−	0.920124i	$-0.628087\pi$
$571$	−10687.0	−0.783252	−0.391626	+	0.920124i	$0.628087\pi$
$572$	4080.00i	0.298240i
$573$	0	0
$574$	6348.00	0.461603
$575$	0	0
$576$	0	0
$577$	− 8329.00i	− 0.600937i	−0.953792	−	0.300469i	$-0.902857\pi$
$577$	− 8329.00i	− 0.600937i	0.953792	−	0.300469i	$-0.0971431\pi$
$578$	6298.00i	0.453222i
$579$	0	0
$580$	0	0
$581$	−25254.0	−1.80329
$582$	0	0
$583$	9900.00i	0.703287i
$584$	5528.00	0.391696
$585$	0	0
$586$	−2976.00	−0.209791
$587$	25302.0i	1.77909i	0.456848	+	0.889545i	$0.348978\pi$
$587$	25302.0i	1.77909i	−0.456848	+	0.889545i	$0.651022\pi$
$588$	0	0
$589$	−7645.00	−0.534816
$590$	0	0
$591$	0	0
$592$	3056.00i	0.212164i
$593$	− 22248.0i	− 1.54067i	−0.637641	−	0.770334i	$-0.720090\pi$
$593$	− 22248.0i	− 1.54067i	0.637641	−	0.770334i	$-0.279910\pi$
$594$	0	0
$595$	0	0
$596$	10008.0	0.687825
$597$	0	0
$598$	− 13056.0i	− 0.892809i
$599$	24252.0	1.65427	0.827137	−	0.562001i	$-0.189968\pi$
$599$	24252.0	1.65427	0.827137	+	0.562001i	$0.189968\pi$
$600$	0	0
$601$	−9829.00	−0.667110	−0.333555	−	0.942731i	$-0.608248\pi$
$601$	−9829.00	−0.667110	−0.333555	+	0.942731i	$0.608248\pi$
$602$	2438.00i	0.165059i
$603$	0	0
$604$	11068.0	0.745613
$605$	0	0
$606$	0	0
$607$	− 14155.0i	− 0.946514i	−0.880925	−	0.473257i	$-0.843078\pi$
$607$	− 14155.0i	− 0.946514i	0.880925	−	0.473257i	$-0.156922\pi$
$608$	4448.00i	0.296694i
$609$	0	0
$610$	0	0
$611$	12444.0	0.823945
$612$	0	0
$613$	− 23051.0i	− 1.51879i	−0.650627	−	0.759397i	$-0.725494\pi$
$613$	− 23051.0i	− 1.51879i	0.650627	−	0.759397i	$-0.274506\pi$
$614$	−14974.0	−0.984204
$615$	0	0
$616$	−5520.00	−0.361051
$617$	8352.00i	0.544958i	0.962162	+	0.272479i	$0.0878435\pi$
$617$	8352.00i	0.544958i	−0.962162	+	0.272479i	$0.912157\pi$
$618$	0	0
$619$	24331.0	1.57988	0.789940	−	0.613184i	$-0.210112\pi$
$619$	24331.0	1.57988	0.789940	+	0.613184i	$0.210112\pi$
$620$	0	0
$621$	0	0
$622$	− 16236.0i	− 1.04663i
$623$	− 18768.0i	− 1.20694i
$624$	0	0
$625$	0	0
$626$	−10004.0	−0.638722
$627$	0	0
$628$	10804.0i	0.686507i
$629$	−8022.00	−0.508518
$630$	0	0
$631$	2216.00	0.139806	0.0699030	−	0.997554i	$-0.477731\pi$
$631$	2216.00	0.139806	0.0699030	+	0.997554i	$0.477731\pi$
$632$	− 5672.00i	− 0.356994i
$633$	0	0
$634$	10164.0	0.636694
$635$	0	0
$636$	0	0
$637$	− 6324.00i	− 0.393353i
$638$	− 14040.0i	− 0.871237i
$639$	0	0
$640$	0	0
$641$	28500.0	1.75613	0.878067	−	0.478537i	$-0.158833\pi$
$641$	28500.0	1.75613	0.878067	+	0.478537i	$0.158833\pi$
$642$	0	0
$643$	25252.0i	1.54874i	0.632731	+	0.774371i	$0.281934\pi$
$643$	25252.0i	1.54874i	−0.632731	+	0.774371i	$0.718066\pi$
$644$	17664.0	1.08084
$645$	0	0
$646$	−11676.0	−0.711124
$647$	28920.0i	1.75728i	0.477481	+	0.878642i	$0.341550\pi$
$647$	28920.0i	1.75728i	−0.477481	+	0.878642i	$0.658450\pi$
$648$	0	0
$649$	11880.0	0.718537
$650$	0	0
$651$	0	0
$652$	6992.00i	0.419981i
$653$	32268.0i	1.93376i	0.255234	+	0.966879i	$0.417848\pi$
$653$	32268.0i	1.93376i	−0.255234	+	0.966879i	$0.582152\pi$
$654$	0	0
$655$	0	0
$656$	−2208.00	−0.131415
$657$	0	0
$658$	16836.0i	0.997471i
$659$	−10050.0	−0.594070	−0.297035	−	0.954867i	$-0.595998\pi$
$659$	−10050.0	−0.594070	−0.297035	+	0.954867i	$0.595998\pi$
$660$	0	0
$661$	−4561.00	−0.268385	−0.134192	−	0.990955i	$-0.542844\pi$
$661$	−4561.00	−0.268385	−0.134192	+	0.990955i	$0.542844\pi$
$662$	15250.0i	0.895329i
$663$	0	0
$664$	8784.00	0.513381
$665$	0	0
$666$	0	0
$667$	44928.0i	2.60812i
$668$	− 2136.00i	− 0.123719i
$669$	0	0
$670$	0	0
$671$	−690.000	−0.0396977
$672$	0	0
$673$	− 21359.0i	− 1.22337i	−0.791101	−	0.611686i	$-0.790492\pi$
$673$	− 21359.0i	− 1.22337i	0.791101	−	0.611686i	$-0.209508\pi$
$674$	7556.00	0.431819
$675$	0	0
$676$	−4164.00	−0.236914
$677$	− 15042.0i	− 0.853931i	−0.904268	−	0.426965i	$-0.859583\pi$
$677$	− 15042.0i	− 0.853931i	0.904268	−	0.426965i	$-0.140417\pi$
$678$	0	0
$679$	−20815.0	−1.17645
$680$	0	0
$681$	0	0
$682$	3300.00i	0.185284i
$683$	27462.0i	1.53851i	0.638940	+	0.769256i	$0.279373\pi$
$683$	27462.0i	1.53851i	−0.638940	+	0.769256i	$0.720627\pi$
$684$	0	0
$685$	0	0
$686$	−7222.00	−0.401949
$687$	0	0
$688$	− 848.000i	− 0.0469908i
$689$	11220.0	0.620389
$690$	0	0
$691$	6212.00	0.341991	0.170995	−	0.985272i	$-0.445302\pi$
$691$	6212.00	0.341991	0.170995	+	0.985272i	$0.445302\pi$
$692$	768.000i	0.0421893i
$693$	0	0
$694$	−16536.0	−0.904464
$695$	0	0
$696$	0	0
$697$	− 5796.00i	− 0.314977i
$698$	− 2758.00i	− 0.149559i
$699$	0	0
$700$	0	0
$701$	13224.0	0.712502	0.356251	−	0.934390i	$-0.384055\pi$
$701$	13224.0	0.712502	0.356251	+	0.934390i	$0.384055\pi$
$702$	0	0
$703$	26549.0i	1.42434i
$704$	1920.00	0.102788
$705$	0	0
$706$	−6144.00	−0.327525
$707$	29394.0i	1.56361i
$708$	0	0
$709$	−34709.0	−1.83854	−0.919269	−	0.393629i	$-0.871219\pi$
$709$	−34709.0	−1.83854	−0.919269	+	0.393629i	$0.871219\pi$
$710$	0	0
$711$	0	0
$712$	6528.00i	0.343606i
$713$	− 10560.0i	− 0.554664i
$714$	0	0
$715$	0	0
$716$	−4560.00	−0.238010
$717$	0	0
$718$	− 14892.0i	− 0.774045i
$719$	32070.0	1.66343	0.831717	−	0.555200i	$-0.187358\pi$
$719$	32070.0	1.66343	0.831717	+	0.555200i	$0.187358\pi$
$720$	0	0
$721$	13915.0	0.718754
$722$	24924.0i	1.28473i
$723$	0	0
$724$	−1592.00	−0.0817213
$725$	0	0
$726$	0	0
$727$	125.000i	0.00637688i	0.999995	+	0.00318844i	$0.00101491\pi$
$727$	125.000i	0.00637688i	−0.999995	+	0.00318844i	$0.998985\pi$
$728$	6256.00i	0.318493i
$729$	0	0
$730$	0	0
$731$	2226.00	0.112629
$732$	0	0
$733$	− 32222.0i	− 1.62367i	−0.583890	−	0.811833i	$-0.698470\pi$
$733$	− 32222.0i	− 1.62367i	0.583890	−	0.811833i	$-0.301530\pi$
$734$	12992.0	0.653329
$735$	0	0
$736$	−6144.00	−0.307705
$737$	− 13560.0i	− 0.677733i
$738$	0	0
$739$	19240.0	0.957720	0.478860	−	0.877891i	$-0.341050\pi$
$739$	19240.0	0.957720	0.478860	+	0.877891i	$0.341050\pi$
$740$	0	0
$741$	0	0
$742$	15180.0i	0.751045i
$743$	30252.0i	1.49373i	0.664978	+	0.746863i	$0.268441\pi$
$743$	30252.0i	1.49373i	−0.664978	+	0.746863i	$0.731559\pi$
$744$	0	0
$745$	0	0
$746$	−3266.00	−0.160291
$747$	0	0
$748$	5040.00i	0.246365i
$749$	34224.0	1.66958
$750$	0	0
$751$	−12517.0	−0.608192	−0.304096	−	0.952641i	$-0.598354\pi$
$751$	−12517.0	−0.608192	−0.304096	+	0.952641i	$0.598354\pi$
$752$	− 5856.00i	− 0.283971i
$753$	0	0
$754$	−15912.0	−0.768542
$755$	0	0
$756$	0	0
$757$	2201.00i	0.105676i	0.998603	+	0.0528380i	$0.0168267\pi$
$757$	2201.00i	0.105676i	−0.998603	+	0.0528380i	$0.983173\pi$
$758$	− 13576.0i	− 0.650531i
$759$	0	0
$760$	0	0
$761$	−8742.00	−0.416422	−0.208211	−	0.978084i	$-0.566764\pi$
$761$	−8742.00	−0.416422	−0.208211	+	0.978084i	$0.566764\pi$
$762$	0	0
$763$	− 13639.0i	− 0.647136i
$764$	13896.0	0.658036
$765$	0	0
$766$	−13164.0	−0.620933
$767$	− 13464.0i	− 0.633842i
$768$	0	0
$769$	28618.0	1.34199	0.670996	−	0.741461i	$-0.265867\pi$
$769$	28618.0	1.34199	0.670996	+	0.741461i	$0.265867\pi$
$770$	0	0
$771$	0	0
$772$	− 10852.0i	− 0.505922i
$773$	6594.00i	0.306817i	0.988163	+	0.153409i	$0.0490251\pi$
$773$	6594.00i	0.306817i	−0.988163	+	0.153409i	$0.950975\pi$
$774$	0	0
$775$	0	0
$776$	7240.00	0.334924
$777$	0	0
$778$	5700.00i	0.262667i
$779$	−19182.0	−0.882242
$780$	0	0
$781$	6120.00	0.280398
$782$	− 16128.0i	− 0.737514i
$783$	0	0
$784$	−2976.00	−0.135569
$785$	0	0
$786$	0	0
$787$	15881.0i	0.719309i	0.933085	+	0.359655i	$0.117106\pi$
$787$	15881.0i	0.719309i	−0.933085	+	0.359655i	$0.882894\pi$
$788$	18936.0i	0.856050i
$789$	0	0
$790$	0	0
$791$	−7452.00	−0.334972
$792$	0	0
$793$	782.000i	0.0350185i
$794$	−14902.0	−0.666061
$795$	0	0
$796$	20528.0	0.914065
$797$	26052.0i	1.15785i	0.815380	+	0.578927i	$0.196528\pi$
$797$	26052.0i	1.15785i	−0.815380	+	0.578927i	$0.803472\pi$
$798$	0	0
$799$	15372.0	0.680629
$800$	0	0
$801$	0	0
$802$	− 28248.0i	− 1.24373i
$803$	− 20730.0i	− 0.911016i
$804$	0	0
$805$	0	0
$806$	3740.00	0.163444
$807$	0	0
$808$	− 10224.0i	− 0.445147i
$809$	12648.0	0.549666	0.274833	−	0.961492i	$-0.411377\pi$
$809$	12648.0	0.549666	0.274833	+	0.961492i	$0.411377\pi$
$810$	0	0
$811$	27179.0	1.17680	0.588399	−	0.808570i	$-0.299758\pi$
$811$	27179.0	1.17680	0.588399	+	0.808570i	$0.299758\pi$
$812$	− 21528.0i	− 0.930400i
$813$	0	0
$814$	11460.0	0.493456
$815$	0	0
$816$	0	0
$817$	− 7367.00i	− 0.315470i
$818$	− 12748.0i	− 0.544894i
$819$	0	0
$820$	0	0
$821$	−11874.0	−0.504757	−0.252378	−	0.967629i	$-0.581213\pi$
$821$	−11874.0	−0.504757	−0.252378	+	0.967629i	$0.581213\pi$
$822$	0	0
$823$	18448.0i	0.781357i	0.920527	+	0.390679i	$0.127760\pi$
$823$	18448.0i	0.781357i	−0.920527	+	0.390679i	$0.872240\pi$
$824$	−4840.00	−0.204623
$825$	0	0
$826$	18216.0	0.767331
$827$	3234.00i	0.135982i	0.997686	+	0.0679911i	$0.0216589\pi$
$827$	3234.00i	0.135982i	−0.997686	+	0.0679911i	$0.978341\pi$
$828$	0	0
$829$	32155.0	1.34715	0.673576	−	0.739118i	$-0.264757\pi$
$829$	32155.0	1.34715	0.673576	+	0.739118i	$0.264757\pi$
$830$	0	0
$831$	0	0
$832$	− 2176.00i	− 0.0906721i
$833$	− 7812.00i	− 0.324934i
$834$	0	0
$835$	0	0
$836$	16680.0	0.690060
$837$	0	0
$838$	7896.00i	0.325493i
$839$	−21996.0	−0.905109	−0.452554	−	0.891737i	$-0.649487\pi$
$839$	−21996.0	−0.905109	−0.452554	+	0.891737i	$0.649487\pi$
$840$	0	0
$841$	30367.0	1.24511
$842$	25258.0i	1.03379i
$843$	0	0
$844$	−20960.0	−0.854826
$845$	0	0
$846$	0	0
$847$	− 9913.00i	− 0.402143i
$848$	− 5280.00i	− 0.213816i
$849$	0	0
$850$	0	0
$851$	−36672.0	−1.47720
$852$	0	0
$853$	− 3278.00i	− 0.131579i	−0.997834	−	0.0657893i	$-0.979043\pi$
$853$	− 3278.00i	− 0.131579i	0.997834	−	0.0657893i	$-0.0209565\pi$
$854$	−1058.00	−0.0423935
$855$	0	0
$856$	−11904.0	−0.475316
$857$	21228.0i	0.846131i	0.906099	+	0.423066i	$0.139046\pi$
$857$	21228.0i	0.846131i	−0.906099	+	0.423066i	$0.860954\pi$
$858$	0	0
$859$	5767.00	0.229066	0.114533	−	0.993419i	$-0.463463\pi$
$859$	5767.00	0.229066	0.114533	+	0.993419i	$0.463463\pi$
$860$	0	0
$861$	0	0
$862$	− 9684.00i	− 0.382643i
$863$	− 5322.00i	− 0.209922i	−0.994476	−	0.104961i	$-0.966528\pi$
$863$	− 5322.00i	− 0.209922i	0.994476	−	0.104961i	$-0.0334718\pi$
$864$	0	0
$865$	0	0
$866$	7702.00	0.302222
$867$	0	0
$868$	5060.00i	0.197866i
$869$	−21270.0	−0.830305
$870$	0	0
$871$	−15368.0	−0.597847
$872$	4744.00i	0.184234i
$873$	0	0
$874$	−53376.0	−2.06576
$875$	0	0
$876$	0	0
$877$	− 42097.0i	− 1.62088i	−0.585819	−	0.810442i	$-0.699227\pi$
$877$	− 42097.0i	− 1.62088i	0.585819	−	0.810442i	$-0.300773\pi$
$878$	14870.0i	0.571570i
$879$	0	0
$880$	0	0
$881$	17658.0	0.675270	0.337635	−	0.941277i	$-0.390373\pi$
$881$	17658.0	0.675270	0.337635	+	0.941277i	$0.390373\pi$
$882$	0	0
$883$	22297.0i	0.849778i	0.905246	+	0.424889i	$0.139687\pi$
$883$	22297.0i	0.849778i	−0.905246	+	0.424889i	$0.860313\pi$
$884$	5712.00	0.217325
$885$	0	0
$886$	11520.0	0.436819
$887$	10542.0i	0.399059i	0.979892	+	0.199530i	$0.0639414\pi$
$887$	10542.0i	0.399059i	−0.979892	+	0.199530i	$0.936059\pi$
$888$	0	0
$889$	−44344.0	−1.67295
$890$	0	0
$891$	0	0
$892$	− 18076.0i	− 0.678508i
$893$	− 50874.0i	− 1.90642i
$894$	0	0
$895$	0	0
$896$	2944.00	0.109768
$897$	0	0
$898$	− 4380.00i	− 0.162764i
$899$	−12870.0	−0.477462
$900$	0	0
$901$	13860.0	0.512479
$902$	8280.00i	0.305647i
$903$	0	0
$904$	2592.00	0.0953635
$905$	0	0
$906$	0	0
$907$	47639.0i	1.74402i	0.489487	+	0.872010i	$0.337184\pi$
$907$	47639.0i	1.74402i	−0.489487	+	0.872010i	$0.662816\pi$
$908$	20256.0i	0.740329i
$909$	0	0
$910$	0	0
$911$	10326.0	0.375539	0.187769	−	0.982213i	$-0.439874\pi$
$911$	10326.0	0.375539	0.187769	+	0.982213i	$0.439874\pi$
$912$	0	0
$913$	− 32940.0i	− 1.19404i
$914$	−14404.0	−0.521271
$915$	0	0
$916$	10292.0	0.371242
$917$	− 63066.0i	− 2.27113i
$918$	0	0
$919$	−5147.00	−0.184748	−0.0923742	−	0.995724i	$-0.529446\pi$
$919$	−5147.00	−0.184748	−0.0923742	+	0.995724i	$0.529446\pi$
$920$	0	0
$921$	0	0
$922$	26952.0i	0.962708i
$923$	− 6936.00i	− 0.247347i
$924$	0	0
$925$	0	0
$926$	−21686.0	−0.769596
$927$	0	0
$928$	7488.00i	0.264877i
$929$	−47064.0	−1.66213	−0.831066	−	0.556175i	$-0.812269\pi$
$929$	−47064.0	−1.66213	−0.831066	+	0.556175i	$0.812269\pi$
$930$	0	0
$931$	−25854.0	−0.910130
$932$	− 16392.0i	− 0.576114i
$933$	0	0
$934$	12216.0	0.427965
$935$	0	0
$936$	0	0
$937$	− 15385.0i	− 0.536399i	−0.963363	−	0.268200i	$-0.913571\pi$
$937$	− 15385.0i	− 0.536399i	0.963363	−	0.268200i	$-0.0864287\pi$
$938$	− 20792.0i	− 0.723756i
$939$	0	0
$940$	0	0
$941$	34914.0	1.20953	0.604763	−	0.796406i	$-0.293268\pi$
$941$	34914.0	1.20953	0.604763	+	0.796406i	$0.293268\pi$
$942$	0	0
$943$	− 26496.0i	− 0.914982i
$944$	−6336.00	−0.218453
$945$	0	0
$946$	−3180.00	−0.109293
$947$	− 37434.0i	− 1.28452i	−0.766486	−	0.642261i	$-0.777997\pi$
$947$	− 37434.0i	− 1.28452i	0.766486	−	0.642261i	$-0.222003\pi$
$948$	0	0
$949$	−23494.0	−0.803633
$950$	0	0
$951$	0	0
$952$	7728.00i	0.263094i
$953$	8778.00i	0.298371i	0.988809	+	0.149185i	$0.0476651\pi$
$953$	8778.00i	0.298371i	−0.988809	+	0.149185i	$0.952335\pi$
$954$	0	0
$955$	0	0
$956$	1224.00	0.0414090
$957$	0	0
$958$	19704.0i	0.664517i
$959$	30498.0	1.02694
$960$	0	0
$961$	−26766.0	−0.898459
$962$	− 12988.0i	− 0.435291i
$963$	0	0
$964$	−25928.0	−0.866270
$965$	0	0
$966$	0	0
$967$	− 54061.0i	− 1.79781i	−0.438141	−	0.898906i	$-0.644363\pi$
$967$	− 54061.0i	− 1.79781i	0.438141	−	0.898906i	$-0.355637\pi$
$968$	3448.00i	0.114486i
$969$	0	0
$970$	0	0
$971$	3084.00	0.101926	0.0509631	−	0.998701i	$-0.483771\pi$
$971$	3084.00	0.101926	0.0509631	+	0.998701i	$0.483771\pi$
$972$	0	0
$973$	− 20539.0i	− 0.676722i
$974$	−15592.0	−0.512936
$975$	0	0
$976$	368.000	0.0120691
$977$	− 12048.0i	− 0.394524i	−0.980351	−	0.197262i	$-0.936795\pi$
$977$	− 12048.0i	− 0.394524i	0.980351	−	0.197262i	$-0.0632049\pi$
$978$	0	0
$979$	24480.0	0.799167
$980$	0	0
$981$	0	0
$982$	− 4908.00i	− 0.159491i
$983$	33618.0i	1.09079i	0.838179	+	0.545396i	$0.183621\pi$
$983$	33618.0i	1.09079i	−0.838179	+	0.545396i	$0.816379\pi$
$984$	0	0
$985$	0	0
$986$	−19656.0	−0.634863
$987$	0	0
$988$	− 18904.0i	− 0.608721i
$989$	10176.0	0.327177
$990$	0	0
$991$	25043.0	0.802742	0.401371	−	0.915916i	$-0.368534\pi$
$991$	25043.0	0.802742	0.401371	+	0.915916i	$0.368534\pi$
$992$	− 1760.00i	− 0.0563307i
$993$	0	0
$994$	9384.00	0.299439
$995$	0	0
$996$	0	0
$997$	− 17710.0i	− 0.562569i	−0.959624	−	0.281285i	$-0.909240\pi$
$997$	− 17710.0i	− 0.562569i	0.959624	−	0.281285i	$-0.0907605\pi$
$998$	5906.00i	0.187326i
$999$	0	0

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1350.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.00000i q^{2} -4.00000 q^{4} +23.0000i q^{7} -8.00000i q^{8} +O(q^{10})\)	\(q+2.00000i q^{2} -4.00000 q^{4} +23.0000i q^{7} -8.00000i q^{8} -30.0000 q^{11} +34.0000i q^{13} -46.0000 q^{14} +16.0000 q^{16} +42.0000i q^{17} +139.000 q^{19} -60.0000i q^{22} +192.000i q^{23} -68.0000 q^{26} -92.0000i q^{28} +234.000 q^{29} -55.0000 q^{31} +32.0000i q^{32} -84.0000 q^{34} +191.000i q^{37} +278.000i q^{38} -138.000 q^{41} -53.0000i q^{43} +120.000 q^{44} -384.000 q^{46} -366.000i q^{47} -186.000 q^{49} -136.000i q^{52} -330.000i q^{53} +184.000 q^{56} +468.000i q^{58} -396.000 q^{59} +23.0000 q^{61} -110.000i q^{62} -64.0000 q^{64} +452.000i q^{67} -168.000i q^{68} -204.000 q^{71} +691.000i q^{73} -382.000 q^{74} -556.000 q^{76} -690.000i q^{77} +709.000 q^{79} -276.000i q^{82} +1098.00i q^{83} +106.000 q^{86} +240.000i q^{88} -816.000 q^{89} -782.000 q^{91} -768.000i q^{92} +732.000 q^{94} +905.000i q^{97} -372.000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{4}+O(q^{10}) \) 2 * q - 8 * q^4	\( 2 q - 8 q^{4} - 60 q^{11} - 92 q^{14} + 32 q^{16} + 278 q^{19} - 136 q^{26} + 468 q^{29} - 110 q^{31} - 168 q^{34} - 276 q^{41} + 240 q^{44} - 768 q^{46} - 372 q^{49} + 368 q^{56} - 792 q^{59} + 46 q^{61} - 128 q^{64} - 408 q^{71} - 764 q^{74} - 1112 q^{76} + 1418 q^{79} + 212 q^{86} - 1632 q^{89} - 1564 q^{91} + 1464 q^{94}+O(q^{100}) \) 2 * q - 8 * q^4 - 60 * q^11 - 92 * q^14 + 32 * q^16 + 278 * q^19 - 136 * q^26 + 468 * q^29 - 110 * q^31 - 168 * q^34 - 276 * q^41 + 240 * q^44 - 768 * q^46 - 372 * q^49 + 368 * q^56 - 792 * q^59 + 46 * q^61 - 128 * q^64 - 408 * q^71 - 764 * q^74 - 1112 * q^76 + 1418 * q^79 + 212 * q^86 - 1632 * q^89 - 1564 * q^91 + 1464 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more