LMFDB - Embedded newform 2025.2.b.g.649.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2025,2,Mod(649,2025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2025, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2025.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2025.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:	$16.1697064093$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 405)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.g.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205i q^{2} -5.46410 q^{4} +1.26795i q^{7} -9.46410i q^{8} +O(q^{10})$	$q+2.73205i q^{2} -5.46410 q^{4} +1.26795i q^{7} -9.46410i q^{8} -2.26795 q^{11} -5.46410i q^{13} -3.46410 q^{14} +14.9282 q^{16} -0.732051i q^{17} +2.46410 q^{19} -6.19615i q^{22} +3.46410i q^{23} +14.9282 q^{26} -6.92820i q^{28} +7.19615 q^{29} -3.00000 q^{31} +21.8564i q^{32} +2.00000 q^{34} -0.732051i q^{37} +6.73205i q^{38} +3.19615 q^{41} -10.1962i q^{43} +12.3923 q^{44} -9.46410 q^{46} +5.26795i q^{47} +5.39230 q^{49} +29.8564i q^{52} +3.26795i q^{53} +12.0000 q^{56} +19.6603i q^{58} +11.7321 q^{59} +4.00000 q^{61} -8.19615i q^{62} -29.8564 q^{64} +3.46410i q^{67} +4.00000i q^{68} -0.267949 q^{71} +9.66025i q^{73} +2.00000 q^{74} -13.4641 q^{76} -2.87564i q^{77} -8.53590 q^{79} +8.73205i q^{82} -8.19615i q^{83} +27.8564 q^{86} +21.4641i q^{88} +5.19615 q^{89} +6.92820 q^{91} -18.9282i q^{92} -14.3923 q^{94} -7.66025i q^{97} +14.7321i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4}+O(q^{10}) $ 4 * q - 8 * q^4	$ 4 q - 8 q^{4} - 16 q^{11} + 32 q^{16} - 4 q^{19} + 32 q^{26} + 8 q^{29} - 12 q^{31} + 8 q^{34} - 8 q^{41} + 8 q^{44} - 24 q^{46} - 20 q^{49} + 48 q^{56} + 40 q^{59} + 16 q^{61} - 64 q^{64} - 8 q^{71} + 8 q^{74} - 40 q^{76} - 48 q^{79} + 56 q^{86} - 16 q^{94}+O(q^{100}) $ 4 * q - 8 * q^4 - 16 * q^11 + 32 * q^16 - 4 * q^19 + 32 * q^26 + 8 * q^29 - 12 * q^31 + 8 * q^34 - 8 * q^41 + 8 * q^44 - 24 * q^46 - 20 * q^49 + 48 * q^56 + 40 * q^59 + 16 * q^61 - 64 * q^64 - 8 * q^71 + 8 * q^74 - 40 * q^76 - 48 * q^79 + 56 * q^86 - 16 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times$.

$n$	$326$	$1702$
$\chi(n)$	$1$	$-1$

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 $

$2$	2.73205i	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$2$	2.73205i	1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$3$	0	0
$3$	0	0
$4$	−5.46410	−2.73205
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.26795i	0.479240i	0.970867	+	0.239620i	$0.0770228\pi$
$7$	1.26795i	0.479240i	−0.970867	+	0.239620i	$0.922977\pi$
$8$	− 9.46410i	− 3.34607i
$9$	0	0
$10$	0	0
$11$	−2.26795	−0.683812	−0.341906	−	0.939734i	$-0.611073\pi$
$11$	−2.26795	−0.683812	−0.341906	+	0.939734i	$0.611073\pi$
$12$	0	0
$13$	− 5.46410i	− 1.51547i	−0.652563	−	0.757735i	$-0.726306\pi$
$13$	− 5.46410i	− 1.51547i	0.652563	−	0.757735i	$-0.273694\pi$
$14$	−3.46410	−0.925820
$15$	0	0
$16$	14.9282	3.73205
$17$	− 0.732051i	− 0.177548i	−0.996052	−	0.0887742i	$-0.971705\pi$
$17$	− 0.732051i	− 0.177548i	0.996052	−	0.0887742i	$-0.0282950\pi$
$18$	0	0
$19$	2.46410	0.565304	0.282652	−	0.959223i	$-0.408786\pi$
$19$	2.46410	0.565304	0.282652	+	0.959223i	$0.408786\pi$
$20$	0	0
$21$	0	0
$22$	− 6.19615i	− 1.32102i
$23$	3.46410i	0.722315i	0.932505	+	0.361158i	$0.117618\pi$
$23$	3.46410i	0.722315i	−0.932505	+	0.361158i	$0.882382\pi$
$24$	0	0
$25$	0	0
$26$	14.9282	2.92766
$27$	0	0
$28$	− 6.92820i	− 1.30931i
$29$	7.19615	1.33629	0.668146	−	0.744030i	$-0.267088\pi$
$29$	7.19615	1.33629	0.668146	+	0.744030i	$0.267088\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	21.8564i	3.86370i
$33$	0	0
$34$	2.00000	0.342997
$35$	0	0
$36$	0	0
$37$	− 0.732051i	− 0.120348i	−0.998188	−	0.0601742i	$-0.980834\pi$
$37$	− 0.732051i	− 0.120348i	0.998188	−	0.0601742i	$-0.0191656\pi$
$38$	6.73205i	1.09208i
$39$	0	0
$40$	0	0
$41$	3.19615	0.499155	0.249578	−	0.968355i	$-0.419708\pi$
$41$	3.19615	0.499155	0.249578	+	0.968355i	$0.419708\pi$
$42$	0	0
$43$	− 10.1962i	− 1.55490i	−0.628946	−	0.777449i	$-0.716513\pi$
$43$	− 10.1962i	− 1.55490i	0.628946	−	0.777449i	$-0.283487\pi$
$44$	12.3923	1.86821
$45$	0	0
$46$	−9.46410	−1.39541
$47$	5.26795i	0.768409i	0.923248	+	0.384205i	$0.125524\pi$
$47$	5.26795i	0.768409i	−0.923248	+	0.384205i	$0.874476\pi$
$48$	0	0
$49$	5.39230	0.770329
$50$	0	0
$51$	0	0
$52$	29.8564i	4.14034i
$53$	3.26795i	0.448887i	0.974487	+	0.224444i	$0.0720565\pi$
$53$	3.26795i	0.448887i	−0.974487	+	0.224444i	$0.927944\pi$
$54$	0	0
$55$	0	0
$56$	12.0000	1.60357
$57$	0	0
$58$	19.6603i	2.58152i
$59$	11.7321	1.52738	0.763691	−	0.645581i	$-0.223385\pi$
$59$	11.7321	1.52738	0.763691	+	0.645581i	$0.223385\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	− 8.19615i	− 1.04091i
$63$	0	0
$64$	−29.8564	−3.73205
$65$	0	0
$66$	0	0
$67$	3.46410i	0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	3.46410i	0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	4.00000i	0.485071i
$69$	0	0
$70$	0	0
$71$	−0.267949	−0.0317997	−0.0158999	−	0.999874i	$-0.505061\pi$
$71$	−0.267949	−0.0317997	−0.0158999	+	0.999874i	$0.505061\pi$
$72$	0	0
$73$	9.66025i	1.13065i	0.824869	+	0.565324i	$0.191249\pi$
$73$	9.66025i	1.13065i	−0.824869	+	0.565324i	$0.808751\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−13.4641	−1.54444
$77$	− 2.87564i	− 0.327710i
$78$	0	0
$79$	−8.53590	−0.960364	−0.480182	−	0.877169i	$-0.659429\pi$
$79$	−8.53590	−0.960364	−0.480182	+	0.877169i	$0.659429\pi$
$80$	0	0
$81$	0	0
$82$	8.73205i	0.964294i
$83$	− 8.19615i	− 0.899645i	−0.893118	−	0.449822i	$-0.851487\pi$
$83$	− 8.19615i	− 0.899645i	0.893118	−	0.449822i	$-0.148513\pi$
$84$	0	0
$85$	0	0
$86$	27.8564	3.00383
$87$	0	0
$88$	21.4641i	2.28808i
$89$	5.19615	0.550791	0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615	0.550791	0.275396	+	0.961331i	$0.411191\pi$
$90$	0	0
$91$	6.92820	0.726273
$92$	− 18.9282i	− 1.97340i
$93$	0	0
$94$	−14.3923	−1.48445
$95$	0	0
$96$	0	0
$97$	− 7.66025i	− 0.777781i	−0.921284	−	0.388890i	$-0.872858\pi$
$97$	− 7.66025i	− 0.777781i	0.921284	−	0.388890i	$-0.127142\pi$
$98$	14.7321i	1.48816i
$99$	0	0
$100$	0	0
$101$	14.6603	1.45875	0.729375	−	0.684114i	$-0.239811\pi$
$101$	14.6603	1.45875	0.729375	+	0.684114i	$0.239811\pi$
$102$	0	0
$103$	− 7.46410i	− 0.735460i	−0.929933	−	0.367730i	$-0.880135\pi$
$103$	− 7.46410i	− 0.735460i	0.929933	−	0.367730i	$-0.119865\pi$
$104$	−51.7128	−5.07086
$105$	0	0
$106$	−8.92820	−0.867184
$107$	15.4641i	1.49497i	0.664278	+	0.747486i	$0.268739\pi$
$107$	15.4641i	1.49497i	−0.664278	+	0.747486i	$0.731261\pi$
$108$	0	0
$109$	19.9282	1.90878	0.954388	−	0.298570i	$-0.0965095\pi$
$109$	19.9282	1.90878	0.954388	+	0.298570i	$0.0965095\pi$
$110$	0	0
$111$	0	0
$112$	18.9282i	1.78855i
$113$	5.12436i	0.482059i	0.970518	+	0.241029i	$0.0774850\pi$
$113$	5.12436i	0.482059i	−0.970518	+	0.241029i	$0.922515\pi$
$114$	0	0
$115$	0	0
$116$	−39.3205	−3.65082
$117$	0	0
$118$	32.0526i	2.95068i
$119$	0.928203	0.0850883
$120$	0	0
$121$	−5.85641	−0.532401
$122$	10.9282i	0.989393i
$123$	0	0
$124$	16.3923	1.47207
$125$	0	0
$126$	0	0
$127$	− 16.5885i	− 1.47199i	−0.676988	−	0.735994i	$-0.736715\pi$
$127$	− 16.5885i	− 1.47199i	0.676988	−	0.735994i	$-0.263285\pi$
$128$	− 37.8564i	− 3.34607i
$129$	0	0
$130$	0	0
$131$	−15.5885	−1.36197	−0.680985	−	0.732297i	$-0.738448\pi$
$131$	−15.5885	−1.36197	−0.680985	+	0.732297i	$0.738448\pi$
$132$	0	0
$133$	3.12436i	0.270916i
$134$	−9.46410	−0.817574
$135$	0	0
$136$	−6.92820	−0.594089
$137$	− 9.46410i	− 0.808573i	−0.914632	−	0.404286i	$-0.867520\pi$
$137$	− 9.46410i	− 0.808573i	0.914632	−	0.404286i	$-0.132480\pi$
$138$	0	0
$139$	−21.3923	−1.81447	−0.907236	−	0.420622i	$-0.861812\pi$
$139$	−21.3923	−1.81447	−0.907236	+	0.420622i	$0.861812\pi$
$140$	0	0
$141$	0	0
$142$	− 0.732051i	− 0.0614323i
$143$	12.3923i	1.03630i
$144$	0	0
$145$	0	0
$146$	−26.3923	−2.18424
$147$	0	0
$148$	4.00000i	0.328798i
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	−15.3923	−1.25261	−0.626304	−	0.779579i	$-0.715433\pi$
$151$	−15.3923	−1.25261	−0.626304	+	0.779579i	$0.715433\pi$
$152$	− 23.3205i	− 1.89154i
$153$	0	0
$154$	7.85641	0.633087
$155$	0	0
$156$	0	0
$157$	− 5.12436i	− 0.408968i	−0.978870	−	0.204484i	$-0.934448\pi$
$157$	− 5.12436i	− 0.408968i	0.978870	−	0.204484i	$-0.0655517\pi$
$158$	− 23.3205i	− 1.85528i
$159$	0	0
$160$	0	0
$161$	−4.39230	−0.346162
$162$	0	0
$163$	9.26795i	0.725922i	0.931804	+	0.362961i	$0.118234\pi$
$163$	9.26795i	0.725922i	−0.931804	+	0.362961i	$0.881766\pi$
$164$	−17.4641	−1.36372
$165$	0	0
$166$	22.3923	1.73798
$167$	0.339746i	0.0262903i	0.999914	+	0.0131452i	$0.00418436\pi$
$167$	0.339746i	0.0262903i	−0.999914	+	0.0131452i	$0.995816\pi$
$168$	0	0
$169$	−16.8564	−1.29665
$170$	0	0
$171$	0	0
$172$	55.7128i	4.24806i
$173$	15.4641i	1.17571i	0.808965	+	0.587857i	$0.200028\pi$
$173$	15.4641i	1.17571i	−0.808965	+	0.587857i	$0.799972\pi$
$174$	0	0
$175$	0	0
$176$	−33.8564	−2.55202
$177$	0	0
$178$	14.1962i	1.06405i
$179$	16.1244	1.20519	0.602595	−	0.798047i	$-0.294133\pi$
$179$	16.1244	1.20519	0.602595	+	0.798047i	$0.294133\pi$
$180$	0	0
$181$	19.5359	1.45209	0.726046	−	0.687646i	$-0.241356\pi$
$181$	19.5359	1.45209	0.726046	+	0.687646i	$0.241356\pi$
$182$	18.9282i	1.40305i
$183$	0	0
$184$	32.7846	2.41691
$185$	0	0
$186$	0	0
$187$	1.66025i	0.121410i
$188$	− 28.7846i	− 2.09933i
$189$	0	0
$190$	0	0
$191$	16.1244	1.16672	0.583359	−	0.812215i	$-0.301738\pi$
$191$	16.1244	1.16672	0.583359	+	0.812215i	$0.301738\pi$
$192$	0	0
$193$	− 8.73205i	− 0.628547i	−0.949333	−	0.314273i	$-0.898239\pi$
$193$	− 8.73205i	− 0.628547i	0.949333	−	0.314273i	$-0.101761\pi$
$194$	20.9282	1.50256
$195$	0	0
$196$	−29.4641	−2.10458
$197$	− 13.8564i	− 0.987228i	−0.869681	−	0.493614i	$-0.835676\pi$
$197$	− 13.8564i	− 0.987228i	0.869681	−	0.493614i	$-0.164324\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	0	0
$202$	40.0526i	2.81809i
$203$	9.12436i	0.640404i
$204$	0	0
$205$	0	0
$206$	20.3923	1.42080
$207$	0	0
$208$	− 81.5692i	− 5.65581i
$209$	−5.58846	−0.386562
$210$	0	0
$211$	18.8564	1.29813	0.649064	−	0.760734i	$-0.275161\pi$
$211$	18.8564	1.29813	0.649064	+	0.760734i	$0.275161\pi$
$212$	− 17.8564i	− 1.22638i
$213$	0	0
$214$	−42.2487	−2.88806
$215$	0	0
$216$	0	0
$217$	− 3.80385i	− 0.258222i
$218$	54.4449i	3.68747i
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	− 24.7846i	− 1.65970i	−0.557986	−	0.829850i	$-0.688426\pi$
$223$	− 24.7846i	− 1.65970i	0.557986	−	0.829850i	$-0.311574\pi$
$224$	−27.7128	−1.85164
$225$	0	0
$226$	−14.0000	−0.931266
$227$	− 20.0526i	− 1.33094i	−0.746427	−	0.665468i	$-0.768232\pi$
$227$	− 20.0526i	− 1.33094i	0.746427	−	0.665468i	$-0.231768\pi$
$228$	0	0
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	0	0
$231$	0	0
$232$	− 68.1051i	− 4.47132i
$233$	10.0526i	0.658565i	0.944231	+	0.329283i	$0.106807\pi$
$233$	10.0526i	0.658565i	−0.944231	+	0.329283i	$0.893193\pi$
$234$	0	0
$235$	0	0
$236$	−64.1051	−4.17289
$237$	0	0
$238$	2.53590i	0.164378i
$239$	7.46410	0.482813	0.241406	−	0.970424i	$-0.422391\pi$
$239$	7.46410	0.482813	0.241406	+	0.970424i	$0.422391\pi$
$240$	0	0
$241$	18.3205	1.18013	0.590064	−	0.807357i	$-0.299103\pi$
$241$	18.3205	1.18013	0.590064	+	0.807357i	$0.299103\pi$
$242$	− 16.0000i	− 1.02852i
$243$	0	0
$244$	−21.8564	−1.39921
$245$	0	0
$246$	0	0
$247$	− 13.4641i	− 0.856700i
$248$	28.3923i	1.80291i
$249$	0	0
$250$	0	0
$251$	10.3923	0.655956	0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923	0.655956	0.327978	+	0.944685i	$0.393633\pi$
$252$	0	0
$253$	− 7.85641i	− 0.493928i
$254$	45.3205	2.84366
$255$	0	0
$256$	43.7128	2.73205
$257$	10.3923i	0.648254i	0.946014	+	0.324127i	$0.105071\pi$
$257$	10.3923i	0.648254i	−0.946014	+	0.324127i	$0.894929\pi$
$258$	0	0
$259$	0.928203	0.0576757
$260$	0	0
$261$	0	0
$262$	− 42.5885i	− 2.63112i
$263$	− 13.3205i	− 0.821378i	−0.911776	−	0.410689i	$-0.865288\pi$
$263$	− 13.3205i	− 0.821378i	0.911776	−	0.410689i	$-0.134712\pi$
$264$	0	0
$265$	0	0
$266$	−8.53590	−0.523370
$267$	0	0
$268$	− 18.9282i	− 1.15622i
$269$	6.66025	0.406083	0.203041	−	0.979170i	$-0.434917\pi$
$269$	6.66025	0.406083	0.203041	+	0.979170i	$0.434917\pi$
$270$	0	0
$271$	10.9282	0.663841	0.331921	−	0.943307i	$-0.392303\pi$
$271$	10.9282	0.663841	0.331921	+	0.943307i	$0.392303\pi$
$272$	− 10.9282i	− 0.662620i
$273$	0	0
$274$	25.8564	1.56204
$275$	0	0
$276$	0	0
$277$	− 14.1962i	− 0.852964i	−0.904496	−	0.426482i	$-0.859753\pi$
$277$	− 14.1962i	− 0.852964i	0.904496	−	0.426482i	$-0.140247\pi$
$278$	− 58.4449i	− 3.50529i
$279$	0	0
$280$	0	0
$281$	8.53590	0.509209	0.254605	−	0.967045i	$-0.418055\pi$
$281$	8.53590	0.509209	0.254605	+	0.967045i	$0.418055\pi$
$282$	0	0
$283$	5.32051i	0.316271i	0.987417	+	0.158136i	$0.0505483\pi$
$283$	5.32051i	0.316271i	−0.987417	+	0.158136i	$0.949452\pi$
$284$	1.46410	0.0868784
$285$	0	0
$286$	−33.8564	−2.00197
$287$	4.05256i	0.239215i
$288$	0	0
$289$	16.4641	0.968477
$290$	0	0
$291$	0	0
$292$	− 52.7846i	− 3.08899i
$293$	25.2679i	1.47617i	0.674708	+	0.738085i	$0.264270\pi$
$293$	25.2679i	1.47617i	−0.674708	+	0.738085i	$0.735730\pi$
$294$	0	0
$295$	0	0
$296$	−6.92820	−0.402694
$297$	0	0
$298$	21.8564i	1.26611i
$299$	18.9282	1.09465
$300$	0	0
$301$	12.9282	0.745169
$302$	− 42.0526i	− 2.41985i
$303$	0	0
$304$	36.7846	2.10974
$305$	0	0
$306$	0	0
$307$	24.0526i	1.37275i	0.727247	+	0.686376i	$0.240800\pi$
$307$	24.0526i	1.37275i	−0.727247	+	0.686376i	$0.759200\pi$
$308$	15.7128i	0.895321i
$309$	0	0
$310$	0	0
$311$	16.2679	0.922471	0.461235	−	0.887278i	$-0.347406\pi$
$311$	16.2679	0.922471	0.461235	+	0.887278i	$0.347406\pi$
$312$	0	0
$313$	− 22.9282i	− 1.29598i	−0.761649	−	0.647989i	$-0.775610\pi$
$313$	− 22.9282i	− 1.29598i	0.761649	−	0.647989i	$-0.224390\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	46.6410	2.62376
$317$	4.19615i	0.235679i	0.993033	+	0.117840i	$0.0375969\pi$
$317$	4.19615i	0.235679i	−0.993033	+	0.117840i	$0.962403\pi$
$318$	0	0
$319$	−16.3205	−0.913773
$320$	0	0
$321$	0	0
$322$	− 12.0000i	− 0.668734i
$323$	− 1.80385i	− 0.100369i
$324$	0	0
$325$	0	0
$326$	−25.3205	−1.40237
$327$	0	0
$328$	− 30.2487i	− 1.67021i
$329$	−6.67949	−0.368252
$330$	0	0
$331$	6.46410	0.355299	0.177650	−	0.984094i	$-0.443151\pi$
$331$	6.46410	0.355299	0.177650	+	0.984094i	$0.443151\pi$
$332$	44.7846i	2.45787i
$333$	0	0
$334$	−0.928203	−0.0507890
$335$	0	0
$336$	0	0
$337$	7.32051i	0.398773i	0.979921	+	0.199387i	$0.0638950\pi$
$337$	7.32051i	0.398773i	−0.979921	+	0.199387i	$0.936105\pi$
$338$	− 46.0526i	− 2.50493i
$339$	0	0
$340$	0	0
$341$	6.80385	0.368449
$342$	0	0
$343$	15.7128i	0.848412i
$344$	−96.4974	−5.20279
$345$	0	0
$346$	−42.2487	−2.27130
$347$	− 2.58846i	− 0.138956i	−0.997583	−	0.0694778i	$-0.977867\pi$
$347$	− 2.58846i	− 0.138956i	0.997583	−	0.0694778i	$-0.0221333\pi$
$348$	0	0
$349$	−8.85641	−0.474073	−0.237036	−	0.971501i	$-0.576176\pi$
$349$	−8.85641	−0.474073	−0.237036	+	0.971501i	$0.576176\pi$
$350$	0	0
$351$	0	0
$352$	− 49.5692i	− 2.64205i
$353$	− 19.5167i	− 1.03877i	−0.854541	−	0.519384i	$-0.826162\pi$
$353$	− 19.5167i	− 1.03877i	0.854541	−	0.519384i	$-0.173838\pi$
$354$	0	0
$355$	0	0
$356$	−28.3923	−1.50479
$357$	0	0
$358$	44.0526i	2.32825i
$359$	−18.1244	−0.956567	−0.478283	−	0.878206i	$-0.658741\pi$
$359$	−18.1244	−0.956567	−0.478283	+	0.878206i	$0.658741\pi$
$360$	0	0
$361$	−12.9282	−0.680432
$362$	53.3731i	2.80523i
$363$	0	0
$364$	−37.8564	−1.98421
$365$	0	0
$366$	0	0
$367$	31.1769i	1.62742i	0.581270	+	0.813711i	$0.302556\pi$
$367$	31.1769i	1.62742i	−0.581270	+	0.813711i	$0.697444\pi$
$368$	51.7128i	2.69572i
$369$	0	0
$370$	0	0
$371$	−4.14359	−0.215125
$372$	0	0
$373$	− 18.0526i	− 0.934726i	−0.884065	−	0.467363i	$-0.845204\pi$
$373$	− 18.0526i	− 0.934726i	0.884065	−	0.467363i	$-0.154796\pi$
$374$	−4.53590	−0.234546
$375$	0	0
$376$	49.8564	2.57115
$377$	− 39.3205i	− 2.02511i
$378$	0	0
$379$	18.3923	0.944749	0.472375	−	0.881398i	$-0.343397\pi$
$379$	18.3923	0.944749	0.472375	+	0.881398i	$0.343397\pi$
$380$	0	0
$381$	0	0
$382$	44.0526i	2.25392i
$383$	9.46410i	0.483593i	0.970327	+	0.241797i	$0.0777366\pi$
$383$	9.46410i	0.483593i	−0.970327	+	0.241797i	$0.922263\pi$
$384$	0	0
$385$	0	0
$386$	23.8564	1.21426
$387$	0	0
$388$	41.8564i	2.12494i
$389$	−20.5359	−1.04121	−0.520606	−	0.853797i	$-0.674294\pi$
$389$	−20.5359	−1.04121	−0.520606	+	0.853797i	$0.674294\pi$
$390$	0	0
$391$	2.53590	0.128246
$392$	− 51.0333i	− 2.57757i
$393$	0	0
$394$	37.8564	1.90718
$395$	0	0
$396$	0	0
$397$	6.39230i	0.320821i	0.987050	+	0.160410i	$0.0512817\pi$
$397$	6.39230i	0.320821i	−0.987050	+	0.160410i	$0.948718\pi$
$398$	5.46410i	0.273891i
$399$	0	0
$400$	0	0
$401$	11.0718	0.552899	0.276450	−	0.961028i	$-0.410842\pi$
$401$	11.0718	0.552899	0.276450	+	0.961028i	$0.410842\pi$
$402$	0	0
$403$	16.3923i	0.816559i
$404$	−80.1051	−3.98538
$405$	0	0
$406$	−24.9282	−1.23717
$407$	1.66025i	0.0822957i
$408$	0	0
$409$	−9.85641	−0.487368	−0.243684	−	0.969855i	$-0.578356\pi$
$409$	−9.85641	−0.487368	−0.243684	+	0.969855i	$0.578356\pi$
$410$	0	0
$411$	0	0
$412$	40.7846i	2.00931i
$413$	14.8756i	0.731983i
$414$	0	0
$415$	0	0
$416$	119.426	5.85532
$417$	0	0
$418$	− 15.2679i	− 0.746780i
$419$	0.392305	0.0191653	0.00958267	−	0.999954i	$-0.496950\pi$
$419$	0.392305	0.0191653	0.00958267	+	0.999954i	$0.496950\pi$
$420$	0	0
$421$	−7.78461	−0.379399	−0.189699	−	0.981842i	$-0.560751\pi$
$421$	−7.78461	−0.379399	−0.189699	+	0.981842i	$0.560751\pi$
$422$	51.5167i	2.50779i
$423$	0	0
$424$	30.9282	1.50201
$425$	0	0
$426$	0	0
$427$	5.07180i	0.245441i
$428$	− 84.4974i	− 4.08434i
$429$	0	0
$430$	0	0
$431$	−38.6603	−1.86220	−0.931099	−	0.364765i	$-0.881149\pi$
$431$	−38.6603	−1.86220	−0.931099	+	0.364765i	$0.881149\pi$
$432$	0	0
$433$	− 28.5359i	− 1.37135i	−0.727909	−	0.685674i	$-0.759508\pi$
$433$	− 28.5359i	− 1.37135i	0.727909	−	0.685674i	$-0.240492\pi$
$434$	10.3923	0.498847
$435$	0	0
$436$	−108.890	−5.21487
$437$	8.53590i	0.408327i
$438$	0	0
$439$	15.3923	0.734635	0.367317	−	0.930096i	$-0.380276\pi$
$439$	15.3923	0.734635	0.367317	+	0.930096i	$0.380276\pi$
$440$	0	0
$441$	0	0
$442$	− 10.9282i	− 0.519802i
$443$	− 17.6603i	− 0.839064i	−0.907741	−	0.419532i	$-0.862194\pi$
$443$	− 17.6603i	− 0.839064i	0.907741	−	0.419532i	$-0.137806\pi$
$444$	0	0
$445$	0	0
$446$	67.7128	3.20629
$447$	0	0
$448$	− 37.8564i	− 1.78855i
$449$	16.1244	0.760955	0.380478	−	0.924790i	$-0.375760\pi$
$449$	16.1244	0.760955	0.380478	+	0.924790i	$0.375760\pi$
$450$	0	0
$451$	−7.24871	−0.341328
$452$	− 28.0000i	− 1.31701i
$453$	0	0
$454$	54.7846	2.57117
$455$	0	0
$456$	0	0
$457$	0.732051i	0.0342439i	0.999853	+	0.0171219i	$0.00545035\pi$
$457$	0.732051i	0.0342439i	−0.999853	+	0.0171219i	$0.994550\pi$
$458$	32.7846i	1.53192i
$459$	0	0
$460$	0	0
$461$	1.05256	0.0490226	0.0245113	−	0.999700i	$-0.492197\pi$
$461$	1.05256	0.0490226	0.0245113	+	0.999700i	$0.492197\pi$
$462$	0	0
$463$	10.3923i	0.482971i	0.970404	+	0.241486i	$0.0776347\pi$
$463$	10.3923i	0.482971i	−0.970404	+	0.241486i	$0.922365\pi$
$464$	107.426	4.98711
$465$	0	0
$466$	−27.4641	−1.27225
$467$	35.3731i	1.63687i	0.574599	+	0.818435i	$0.305158\pi$
$467$	35.3731i	1.63687i	−0.574599	+	0.818435i	$0.694842\pi$
$468$	0	0
$469$	−4.39230	−0.202818
$470$	0	0
$471$	0	0
$472$	− 111.033i	− 5.11072i
$473$	23.1244i	1.06326i
$474$	0	0
$475$	0	0
$476$	−5.07180	−0.232465
$477$	0	0
$478$	20.3923i	0.932722i
$479$	36.1244	1.65056	0.825282	−	0.564721i	$-0.191016\pi$
$479$	36.1244	1.65056	0.825282	+	0.564721i	$0.191016\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	50.0526i	2.27983i
$483$	0	0
$484$	32.0000	1.45455
$485$	0	0
$486$	0	0
$487$	− 3.60770i	− 0.163480i	−0.996654	−	0.0817401i	$-0.973952\pi$
$487$	− 3.60770i	− 0.163480i	0.996654	−	0.0817401i	$-0.0260478\pi$
$488$	− 37.8564i	− 1.71368i
$489$	0	0
$490$	0	0
$491$	−38.1244	−1.72053	−0.860264	−	0.509849i	$-0.829701\pi$
$491$	−38.1244	−1.72053	−0.860264	+	0.509849i	$0.829701\pi$
$492$	0	0
$493$	− 5.26795i	− 0.237256i
$494$	36.7846	1.65502
$495$	0	0
$496$	−44.7846	−2.01089
$497$	− 0.339746i	− 0.0152397i
$498$	0	0
$499$	10.3205	0.462009	0.231005	−	0.972953i	$-0.425799\pi$
$499$	10.3205	0.462009	0.231005	+	0.972953i	$0.425799\pi$
$500$	0	0
$501$	0	0
$502$	28.3923i	1.26721i
$503$	27.3205i	1.21816i	0.793108	+	0.609081i	$0.208461\pi$
$503$	27.3205i	1.21816i	−0.793108	+	0.609081i	$0.791539\pi$
$504$	0	0
$505$	0	0
$506$	21.4641	0.954196
$507$	0	0
$508$	90.6410i	4.02154i
$509$	−34.7846	−1.54180	−0.770900	−	0.636956i	$-0.780193\pi$
$509$	−34.7846	−1.54180	−0.770900	+	0.636956i	$0.780193\pi$
$510$	0	0
$511$	−12.2487	−0.541851
$512$	43.7128i	1.93185i
$513$	0	0
$514$	−28.3923	−1.25233
$515$	0	0
$516$	0	0
$517$	− 11.9474i	− 0.525448i
$518$	2.53590i	0.111421i
$519$	0	0
$520$	0	0
$521$	12.5359	0.549208	0.274604	−	0.961557i	$-0.411453\pi$
$521$	12.5359	0.549208	0.274604	+	0.961557i	$0.411453\pi$
$522$	0	0
$523$	− 26.2487i	− 1.14778i	−0.818934	−	0.573888i	$-0.805434\pi$
$523$	− 26.2487i	− 1.14778i	0.818934	−	0.573888i	$-0.194566\pi$
$524$	85.1769	3.72097
$525$	0	0
$526$	36.3923	1.58678
$527$	2.19615i	0.0956659i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	0	0
$532$	− 17.0718i	− 0.740156i
$533$	− 17.4641i	− 0.756454i
$534$	0	0
$535$	0	0
$536$	32.7846	1.41608
$537$	0	0
$538$	18.1962i	0.784492i
$539$	−12.2295	−0.526761
$540$	0	0
$541$	−17.5359	−0.753927	−0.376964	−	0.926228i	$-0.623032\pi$
$541$	−17.5359	−0.753927	−0.376964	+	0.926228i	$0.623032\pi$
$542$	29.8564i	1.28244i
$543$	0	0
$544$	16.0000	0.685994
$545$	0	0
$546$	0	0
$547$	6.14359i	0.262681i	0.991337	+	0.131341i	$0.0419281\pi$
$547$	6.14359i	0.262681i	−0.991337	+	0.131341i	$0.958072\pi$
$548$	51.7128i	2.20906i
$549$	0	0
$550$	0	0
$551$	17.7321	0.755411
$552$	0	0
$553$	− 10.8231i	− 0.460244i
$554$	38.7846	1.64780
$555$	0	0
$556$	116.890	4.95723
$557$	9.46410i	0.401007i	0.979693	+	0.200503i	$0.0642578\pi$
$557$	9.46410i	0.401007i	−0.979693	+	0.200503i	$0.935742\pi$
$558$	0	0
$559$	−55.7128	−2.35640
$560$	0	0
$561$	0	0
$562$	23.3205i	0.983716i
$563$	− 13.2679i	− 0.559177i	−0.960120	−	0.279589i	$-0.909802\pi$
$563$	− 13.2679i	− 0.559177i	0.960120	−	0.279589i	$-0.0901981\pi$
$564$	0	0
$565$	0	0
$566$	−14.5359	−0.610989
$567$	0	0
$568$	2.53590i	0.106404i
$569$	−32.9090	−1.37962	−0.689808	−	0.723993i	$-0.742305\pi$
$569$	−32.9090	−1.37962	−0.689808	+	0.723993i	$0.742305\pi$
$570$	0	0
$571$	1.78461	0.0746836	0.0373418	−	0.999303i	$-0.488111\pi$
$571$	1.78461	0.0746836	0.0373418	+	0.999303i	$0.488111\pi$
$572$	− 67.7128i	− 2.83121i
$573$	0	0
$574$	−11.0718	−0.462128
$575$	0	0
$576$	0	0
$577$	− 18.7321i	− 0.779825i	−0.920852	−	0.389913i	$-0.872505\pi$
$577$	− 18.7321i	− 0.779825i	0.920852	−	0.389913i	$-0.127495\pi$
$578$	44.9808i	1.87095i
$579$	0	0
$580$	0	0
$581$	10.3923	0.431145
$582$	0	0
$583$	− 7.41154i	− 0.306955i
$584$	91.4256	3.78322
$585$	0	0
$586$	−69.0333	−2.85174
$587$	− 5.66025i	− 0.233624i	−0.993154	−	0.116812i	$-0.962733\pi$
$587$	− 5.66025i	− 0.233624i	0.993154	−	0.116812i	$-0.0372674\pi$
$588$	0	0
$589$	−7.39230	−0.304595
$590$	0	0
$591$	0	0
$592$	− 10.9282i	− 0.449146i
$593$	− 27.8564i	− 1.14393i	−0.820280	−	0.571963i	$-0.806182\pi$
$593$	− 27.8564i	− 1.14393i	0.820280	−	0.571963i	$-0.193818\pi$
$594$	0	0
$595$	0	0
$596$	−43.7128	−1.79055
$597$	0	0
$598$	51.7128i	2.11469i
$599$	−16.8038	−0.686587	−0.343293	−	0.939228i	$-0.611542\pi$
$599$	−16.8038	−0.686587	−0.343293	+	0.939228i	$0.611542\pi$
$600$	0	0
$601$	17.2487	0.703590	0.351795	−	0.936077i	$-0.385571\pi$
$601$	17.2487	0.703590	0.351795	+	0.936077i	$0.385571\pi$
$602$	35.3205i	1.43956i
$603$	0	0
$604$	84.1051	3.42219
$605$	0	0
$606$	0	0
$607$	5.80385i	0.235571i	0.993039	+	0.117785i	$0.0375795\pi$
$607$	5.80385i	0.235571i	−0.993039	+	0.117785i	$0.962420\pi$
$608$	53.8564i	2.18417i
$609$	0	0
$610$	0	0
$611$	28.7846	1.16450
$612$	0	0
$613$	− 5.46410i	− 0.220693i	−0.993893	−	0.110346i	$-0.964804\pi$
$613$	− 5.46410i	− 0.220693i	0.993893	−	0.110346i	$-0.0351961\pi$
$614$	−65.7128	−2.65195
$615$	0	0
$616$	−27.2154	−1.09654
$617$	− 6.92820i	− 0.278919i	−0.990228	−	0.139459i	$-0.955464\pi$
$617$	− 6.92820i	− 0.278919i	0.990228	−	0.139459i	$-0.0445365\pi$
$618$	0	0
$619$	−15.8564	−0.637323	−0.318661	−	0.947869i	$-0.603233\pi$
$619$	−15.8564	−0.637323	−0.318661	+	0.947869i	$0.603233\pi$
$620$	0	0
$621$	0	0
$622$	44.4449i	1.78208i
$623$	6.58846i	0.263961i
$624$	0	0
$625$	0	0
$626$	62.6410	2.50364
$627$	0	0
$628$	28.0000i	1.11732i
$629$	−0.535898	−0.0213677
$630$	0	0
$631$	22.7128	0.904183	0.452091	−	0.891972i	$-0.350678\pi$
$631$	22.7128	0.904183	0.452091	+	0.891972i	$0.350678\pi$
$632$	80.7846i	3.21344i
$633$	0	0
$634$	−11.4641	−0.455298
$635$	0	0
$636$	0	0
$637$	− 29.4641i	− 1.16741i
$638$	− 44.5885i	− 1.76527i
$639$	0	0
$640$	0	0
$641$	26.6603	1.05302	0.526508	−	0.850170i	$-0.323501\pi$
$641$	26.6603	1.05302	0.526508	+	0.850170i	$0.323501\pi$
$642$	0	0
$643$	24.3397i	0.959866i	0.877305	+	0.479933i	$0.159339\pi$
$643$	24.3397i	0.959866i	−0.877305	+	0.479933i	$0.840661\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	4.92820	0.193898
$647$	− 4.53590i	− 0.178325i	−0.996017	−	0.0891623i	$-0.971581\pi$
$647$	− 4.53590i	− 0.178325i	0.996017	−	0.0891623i	$-0.0284190\pi$
$648$	0	0
$649$	−26.6077	−1.04444
$650$	0	0
$651$	0	0
$652$	− 50.6410i	− 1.98326i
$653$	− 10.5359i	− 0.412302i	−0.978520	−	0.206151i	$-0.933906\pi$
$653$	− 10.5359i	− 0.412302i	0.978520	−	0.206151i	$-0.0660937\pi$
$654$	0	0
$655$	0	0
$656$	47.7128	1.86287
$657$	0	0
$658$	− 18.2487i	− 0.711409i
$659$	−5.46410	−0.212851	−0.106426	−	0.994321i	$-0.533941\pi$
$659$	−5.46410	−0.212851	−0.106426	+	0.994321i	$0.533941\pi$
$660$	0	0
$661$	−16.3205	−0.634794	−0.317397	−	0.948293i	$-0.602809\pi$
$661$	−16.3205	−0.634794	−0.317397	+	0.948293i	$0.602809\pi$
$662$	17.6603i	0.686385i
$663$	0	0
$664$	−77.5692	−3.01027
$665$	0	0
$666$	0	0
$667$	24.9282i	0.965224i
$668$	− 1.85641i	− 0.0718265i
$669$	0	0
$670$	0	0
$671$	−9.07180	−0.350213
$672$	0	0
$673$	10.3923i	0.400594i	0.979735	+	0.200297i	$0.0641907\pi$
$673$	10.3923i	0.400594i	−0.979735	+	0.200297i	$0.935809\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	92.1051	3.54250
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	9.71281	0.372744
$680$	0	0
$681$	0	0
$682$	18.5885i	0.711789i
$683$	− 40.3923i	− 1.54557i	−0.634669	−	0.772784i	$-0.718863\pi$
$683$	− 40.3923i	− 1.54557i	0.634669	−	0.772784i	$-0.281137\pi$
$684$	0	0
$685$	0	0
$686$	−42.9282	−1.63901
$687$	0	0
$688$	− 152.210i	− 5.80296i
$689$	17.8564	0.680275
$690$	0	0
$691$	37.7128	1.43466	0.717332	−	0.696732i	$-0.245363\pi$
$691$	37.7128	1.43466	0.717332	+	0.696732i	$0.245363\pi$
$692$	− 84.4974i	− 3.21211i
$693$	0	0
$694$	7.07180	0.268442
$695$	0	0
$696$	0	0
$697$	− 2.33975i	− 0.0886242i
$698$	− 24.1962i	− 0.915838i
$699$	0	0
$700$	0	0
$701$	−31.1962	−1.17826	−0.589131	−	0.808037i	$-0.700530\pi$
$701$	−31.1962	−1.17826	−0.589131	+	0.808037i	$0.700530\pi$
$702$	0	0
$703$	− 1.80385i	− 0.0680334i
$704$	67.7128	2.55202
$705$	0	0
$706$	53.3205	2.00674
$707$	18.5885i	0.699091i
$708$	0	0
$709$	29.4641	1.10655	0.553274	−	0.832999i	$-0.313378\pi$
$709$	29.4641	1.10655	0.553274	+	0.832999i	$0.313378\pi$
$710$	0	0
$711$	0	0
$712$	− 49.1769i	− 1.84298i
$713$	− 10.3923i	− 0.389195i
$714$	0	0
$715$	0	0
$716$	−88.1051	−3.29264
$717$	0	0
$718$	− 49.5167i	− 1.84795i
$719$	−39.5885	−1.47640	−0.738200	−	0.674582i	$-0.764324\pi$
$719$	−39.5885	−1.47640	−0.738200	+	0.674582i	$0.764324\pi$
$720$	0	0
$721$	9.46410	0.352462
$722$	− 35.3205i	− 1.31449i
$723$	0	0
$724$	−106.746	−3.96719
$725$	0	0
$726$	0	0
$727$	− 12.3923i	− 0.459605i	−0.973237	−	0.229803i	$-0.926192\pi$
$727$	− 12.3923i	− 0.459605i	0.973237	−	0.229803i	$-0.0738080\pi$
$728$	− 65.5692i	− 2.43016i
$729$	0	0
$730$	0	0
$731$	−7.46410	−0.276070
$732$	0	0
$733$	− 6.78461i	− 0.250595i	−0.992119	−	0.125298i	$-0.960011\pi$
$733$	− 6.78461i	− 0.250595i	0.992119	−	0.125298i	$-0.0399886\pi$
$734$	−85.1769	−3.14394
$735$	0	0
$736$	−75.7128	−2.79081
$737$	− 7.85641i	− 0.289394i
$738$	0	0
$739$	−15.5359	−0.571497	−0.285749	−	0.958305i	$-0.592242\pi$
$739$	−15.5359	−0.571497	−0.285749	+	0.958305i	$0.592242\pi$
$740$	0	0
$741$	0	0
$742$	− 11.3205i	− 0.415589i
$743$	− 45.9090i	− 1.68424i	−0.539293	−	0.842118i	$-0.681308\pi$
$743$	− 45.9090i	− 1.68424i	0.539293	−	0.842118i	$-0.318692\pi$
$744$	0	0
$745$	0	0
$746$	49.3205	1.80575
$747$	0	0
$748$	− 9.07180i	− 0.331698i
$749$	−19.6077	−0.716450
$750$	0	0
$751$	7.21539	0.263293	0.131647	−	0.991297i	$-0.457974\pi$
$751$	7.21539	0.263293	0.131647	+	0.991297i	$0.457974\pi$
$752$	78.6410i	2.86774i
$753$	0	0
$754$	107.426	3.91221
$755$	0	0
$756$	0	0
$757$	− 53.1769i	− 1.93275i	−0.257141	−	0.966374i	$-0.582780\pi$
$757$	− 53.1769i	− 1.93275i	0.257141	−	0.966374i	$-0.417220\pi$
$758$	50.2487i	1.82512i
$759$	0	0
$760$	0	0
$761$	−35.4449	−1.28488	−0.642438	−	0.766338i	$-0.722077\pi$
$761$	−35.4449	−1.28488	−0.642438	+	0.766338i	$0.722077\pi$
$762$	0	0
$763$	25.2679i	0.914761i
$764$	−88.1051	−3.18753
$765$	0	0
$766$	−25.8564	−0.934230
$767$	− 64.1051i	− 2.31470i
$768$	0	0
$769$	16.4641	0.593711	0.296855	−	0.954922i	$-0.404062\pi$
$769$	16.4641	0.593711	0.296855	+	0.954922i	$0.404062\pi$
$770$	0	0
$771$	0	0
$772$	47.7128i	1.71722i
$773$	− 43.5167i	− 1.56519i	−0.622534	−	0.782593i	$-0.713897\pi$
$773$	− 43.5167i	− 1.56519i	0.622534	−	0.782593i	$-0.286103\pi$
$774$	0	0
$775$	0	0
$776$	−72.4974	−2.60251
$777$	0	0
$778$	− 56.1051i	− 2.01147i
$779$	7.87564	0.282174
$780$	0	0
$781$	0.607695	0.0217450
$782$	6.92820i	0.247752i
$783$	0	0
$784$	80.4974	2.87491
$785$	0	0
$786$	0	0
$787$	9.94744i	0.354588i	0.984158	+	0.177294i	$0.0567343\pi$
$787$	9.94744i	0.354588i	−0.984158	+	0.177294i	$0.943266\pi$
$788$	75.7128i	2.69716i
$789$	0	0
$790$	0	0
$791$	−6.49742	−0.231022
$792$	0	0
$793$	− 21.8564i	− 0.776144i
$794$	−17.4641	−0.619778
$795$	0	0
$796$	−10.9282	−0.387340
$797$	36.3923i	1.28908i	0.764570	+	0.644541i	$0.222951\pi$
$797$	36.3923i	1.28908i	−0.764570	+	0.644541i	$0.777049\pi$
$798$	0	0
$799$	3.85641	0.136430
$800$	0	0
$801$	0	0
$802$	30.2487i	1.06812i
$803$	− 21.9090i	− 0.773151i
$804$	0	0
$805$	0	0
$806$	−44.7846	−1.57747
$807$	0	0
$808$	− 138.746i	− 4.88107i
$809$	13.4449	0.472696	0.236348	−	0.971668i	$-0.424049\pi$
$809$	13.4449	0.472696	0.236348	+	0.971668i	$0.424049\pi$
$810$	0	0
$811$	−11.5359	−0.405080	−0.202540	−	0.979274i	$-0.564920\pi$
$811$	−11.5359	−0.405080	−0.202540	+	0.979274i	$0.564920\pi$
$812$	− 49.8564i	− 1.74962i
$813$	0	0
$814$	−4.53590	−0.158983
$815$	0	0
$816$	0	0
$817$	− 25.1244i	− 0.878990i
$818$	− 26.9282i	− 0.941523i
$819$	0	0
$820$	0	0
$821$	−34.2679	−1.19596	−0.597980	−	0.801511i	$-0.704030\pi$
$821$	−34.2679	−1.19596	−0.597980	+	0.801511i	$0.704030\pi$
$822$	0	0
$823$	− 23.8564i	− 0.831582i	−0.909460	−	0.415791i	$-0.863505\pi$
$823$	− 23.8564i	− 0.831582i	0.909460	−	0.415791i	$-0.136495\pi$
$824$	−70.6410	−2.46090
$825$	0	0
$826$	−40.6410	−1.41408
$827$	32.3923i	1.12639i	0.826324	+	0.563195i	$0.190428\pi$
$827$	32.3923i	1.12639i	−0.826324	+	0.563195i	$0.809572\pi$
$828$	0	0
$829$	−17.7846	−0.617685	−0.308843	−	0.951113i	$-0.599942\pi$
$829$	−17.7846	−0.617685	−0.308843	+	0.951113i	$0.599942\pi$
$830$	0	0
$831$	0	0
$832$	163.138i	5.65581i
$833$	− 3.94744i	− 0.136771i
$834$	0	0
$835$	0	0
$836$	30.5359	1.05611
$837$	0	0
$838$	1.07180i	0.0370246i
$839$	−22.8038	−0.787276	−0.393638	−	0.919265i	$-0.628784\pi$
$839$	−22.8038	−0.787276	−0.393638	+	0.919265i	$0.628784\pi$
$840$	0	0
$841$	22.7846	0.785676
$842$	− 21.2679i	− 0.732942i
$843$	0	0
$844$	−103.033	−3.54655
$845$	0	0
$846$	0	0
$847$	− 7.42563i	− 0.255148i
$848$	48.7846i	1.67527i
$849$	0	0
$850$	0	0
$851$	2.53590	0.0869295
$852$	0	0
$853$	55.5167i	1.90085i	0.310947	+	0.950427i	$0.399354\pi$
$853$	55.5167i	1.90085i	−0.310947	+	0.950427i	$0.600646\pi$
$854$	−13.8564	−0.474156
$855$	0	0
$856$	146.354	5.00227
$857$	52.4974i	1.79328i	0.442763	+	0.896639i	$0.353998\pi$
$857$	52.4974i	1.79328i	−0.442763	+	0.896639i	$0.646002\pi$
$858$	0	0
$859$	11.0000	0.375315	0.187658	−	0.982235i	$-0.439910\pi$
$859$	11.0000	0.375315	0.187658	+	0.982235i	$0.439910\pi$
$860$	0	0
$861$	0	0
$862$	− 105.622i	− 3.59749i
$863$	1.12436i	0.0382735i	0.999817	+	0.0191368i	$0.00609179\pi$
$863$	1.12436i	0.0382735i	−0.999817	+	0.0191368i	$0.993908\pi$
$864$	0	0
$865$	0	0
$866$	77.9615	2.64924
$867$	0	0
$868$	20.7846i	0.705476i
$869$	19.3590	0.656709
$870$	0	0
$871$	18.9282	0.641358
$872$	− 188.603i	− 6.38689i
$873$	0	0
$874$	−23.3205	−0.788828
$875$	0	0
$876$	0	0
$877$	− 26.5359i	− 0.896054i	−0.894020	−	0.448027i	$-0.852127\pi$
$877$	− 26.5359i	− 0.896054i	0.894020	−	0.448027i	$-0.147873\pi$
$878$	42.0526i	1.41921i
$879$	0	0
$880$	0	0
$881$	−8.94744	−0.301447	−0.150723	−	0.988576i	$-0.548160\pi$
$881$	−8.94744	−0.301447	−0.150723	+	0.988576i	$0.548160\pi$
$882$	0	0
$883$	− 19.8038i	− 0.666453i	−0.942847	−	0.333226i	$-0.891863\pi$
$883$	− 19.8038i	− 0.666453i	0.942847	−	0.333226i	$-0.108137\pi$
$884$	21.8564	0.735111
$885$	0	0
$886$	48.2487	1.62095
$887$	− 44.1962i	− 1.48396i	−0.670421	−	0.741981i	$-0.733887\pi$
$887$	− 44.1962i	− 1.48396i	0.670421	−	0.741981i	$-0.266113\pi$
$888$	0	0
$889$	21.0333	0.705435
$890$	0	0
$891$	0	0
$892$	135.426i	4.53439i
$893$	12.9808i	0.434385i
$894$	0	0
$895$	0	0
$896$	48.0000	1.60357
$897$	0	0
$898$	44.0526i	1.47005i
$899$	−21.5885	−0.720015
$900$	0	0
$901$	2.39230	0.0796992
$902$	− 19.8038i	− 0.659396i
$903$	0	0
$904$	48.4974	1.61300
$905$	0	0
$906$	0	0
$907$	30.0000i	0.996134i	0.867139	+	0.498067i	$0.165957\pi$
$907$	30.0000i	0.996134i	−0.867139	+	0.498067i	$0.834043\pi$
$908$	109.569i	3.63618i
$909$	0	0
$910$	0	0
$911$	−7.58846	−0.251417	−0.125708	−	0.992067i	$-0.540120\pi$
$911$	−7.58846	−0.251417	−0.125708	+	0.992067i	$0.540120\pi$
$912$	0	0
$913$	18.5885i	0.615188i
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	−65.5692	−2.16647
$917$	− 19.7654i	− 0.652710i
$918$	0	0
$919$	−46.9615	−1.54912	−0.774559	−	0.632502i	$-0.782028\pi$
$919$	−46.9615	−1.54912	−0.774559	+	0.632502i	$0.782028\pi$
$920$	0	0
$921$	0	0
$922$	2.87564i	0.0947043i
$923$	1.46410i	0.0481915i
$924$	0	0
$925$	0	0
$926$	−28.3923	−0.933029
$927$	0	0
$928$	157.282i	5.16304i
$929$	−28.3731	−0.930890	−0.465445	−	0.885077i	$-0.654106\pi$
$929$	−28.3731	−0.930890	−0.465445	+	0.885077i	$0.654106\pi$
$930$	0	0
$931$	13.2872	0.435470
$932$	− 54.9282i	− 1.79923i
$933$	0	0
$934$	−96.6410	−3.16219
$935$	0	0
$936$	0	0
$937$	31.8564i	1.04070i	0.853952	+	0.520352i	$0.174199\pi$
$937$	31.8564i	1.04070i	−0.853952	+	0.520352i	$0.825801\pi$
$938$	− 12.0000i	− 0.391814i
$939$	0	0
$940$	0	0
$941$	−27.1769	−0.885942	−0.442971	−	0.896536i	$-0.646076\pi$
$941$	−27.1769	−0.885942	−0.442971	+	0.896536i	$0.646076\pi$
$942$	0	0
$943$	11.0718i	0.360547i
$944$	175.138	5.70027
$945$	0	0
$946$	−63.1769	−2.05406
$947$	2.28719i	0.0743236i	0.999309	+	0.0371618i	$0.0118317\pi$
$947$	2.28719i	0.0743236i	−0.999309	+	0.0371618i	$0.988168\pi$
$948$	0	0
$949$	52.7846	1.71346
$950$	0	0
$951$	0	0
$952$	− 8.78461i	− 0.284711i
$953$	15.6077i	0.505583i	0.967521	+	0.252791i	$0.0813486\pi$
$953$	15.6077i	0.505583i	−0.967521	+	0.252791i	$0.918651\pi$
$954$	0	0
$955$	0	0
$956$	−40.7846	−1.31907
$957$	0	0
$958$	98.6936i	3.18864i
$959$	12.0000	0.387500
$960$	0	0
$961$	−22.0000	−0.709677
$962$	− 10.9282i	− 0.352339i
$963$	0	0
$964$	−100.105	−3.22417
$965$	0	0
$966$	0	0
$967$	36.1962i	1.16399i	0.813192	+	0.581995i	$0.197728\pi$
$967$	36.1962i	1.16399i	−0.813192	+	0.581995i	$0.802272\pi$
$968$	55.4256i	1.78145i
$969$	0	0
$970$	0	0
$971$	−41.4449	−1.33003	−0.665014	−	0.746830i	$-0.731575\pi$
$971$	−41.4449	−1.33003	−0.665014	+	0.746830i	$0.731575\pi$
$972$	0	0
$973$	− 27.1244i	− 0.869567i
$974$	9.85641	0.315820
$975$	0	0
$976$	59.7128	1.91136
$977$	1.46410i	0.0468408i	0.999726	+	0.0234204i	$0.00745562\pi$
$977$	1.46410i	0.0468408i	−0.999726	+	0.0234204i	$0.992544\pi$
$978$	0	0
$979$	−11.7846	−0.376638
$980$	0	0
$981$	0	0
$982$	− 104.158i	− 3.32381i
$983$	17.4115i	0.555342i	0.960676	+	0.277671i	$0.0895625\pi$
$983$	17.4115i	0.555342i	−0.960676	+	0.277671i	$0.910437\pi$
$984$	0	0
$985$	0	0
$986$	14.3923	0.458344
$987$	0	0
$988$	73.5692i	2.34055i
$989$	35.3205	1.12313
$990$	0	0
$991$	3.14359	0.0998595	0.0499298	−	0.998753i	$-0.484100\pi$
$991$	3.14359	0.0998595	0.0499298	+	0.998753i	$0.484100\pi$
$992$	− 65.5692i	− 2.08182i
$993$	0	0
$994$	0.928203	0.0294408
$995$	0	0
$996$	0	0
$997$	19.5167i	0.618099i	0.951046	+	0.309049i	$0.100011\pi$
$997$	19.5167i	0.618099i	−0.951046	+	0.309049i	$0.899989\pi$
$998$	28.1962i	0.892534i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.g.649.4		4
3.2	odd	2		2025.2.b.h.649.1		4
5.2	odd	4		405.2.a.g.1.1	✓	2
5.3	odd	4		2025.2.a.m.1.2		2
5.4	even	2	inner	2025.2.b.g.649.1		4
15.2	even	4		405.2.a.h.1.2	yes	2
15.8	even	4		2025.2.a.g.1.1		2
15.14	odd	2		2025.2.b.h.649.4		4
20.7	even	4		6480.2.a.br.1.1		2
45.2	even	12		405.2.e.i.271.1		4
45.7	odd	12		405.2.e.l.271.2		4
45.22	odd	12		405.2.e.l.136.2		4
45.32	even	12		405.2.e.i.136.1		4
60.47	odd	4		6480.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.g.1.1	✓	2	5.2	odd	4
405.2.a.h.1.2	yes	2	15.2	even	4
405.2.e.i.136.1		4	45.32	even	12
405.2.e.i.271.1		4	45.2	even	12
405.2.e.l.136.2		4	45.22	odd	12
405.2.e.l.271.2		4	45.7	odd	12
2025.2.a.g.1.1		2	15.8	even	4
2025.2.a.m.1.2		2	5.3	odd	4
2025.2.b.g.649.1		4	5.4	even	2	inner
2025.2.b.g.649.4		4	1.1	even	1	trivial
2025.2.b.h.649.1		4	3.2	odd	2
2025.2.b.h.649.4		4	15.14	odd	2
6480.2.a.bi.1.1		2	60.47	odd	4
6480.2.a.br.1.1		2	20.7	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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.73205i q^{2} -5.46410 q^{4} +1.26795i q^{7} -9.46410i q^{8} +O(q^{10})\)	\(q+2.73205i q^{2} -5.46410 q^{4} +1.26795i q^{7} -9.46410i q^{8} -2.26795 q^{11} -5.46410i q^{13} -3.46410 q^{14} +14.9282 q^{16} -0.732051i q^{17} +2.46410 q^{19} -6.19615i q^{22} +3.46410i q^{23} +14.9282 q^{26} -6.92820i q^{28} +7.19615 q^{29} -3.00000 q^{31} +21.8564i q^{32} +2.00000 q^{34} -0.732051i q^{37} +6.73205i q^{38} +3.19615 q^{41} -10.1962i q^{43} +12.3923 q^{44} -9.46410 q^{46} +5.26795i q^{47} +5.39230 q^{49} +29.8564i q^{52} +3.26795i q^{53} +12.0000 q^{56} +19.6603i q^{58} +11.7321 q^{59} +4.00000 q^{61} -8.19615i q^{62} -29.8564 q^{64} +3.46410i q^{67} +4.00000i q^{68} -0.267949 q^{71} +9.66025i q^{73} +2.00000 q^{74} -13.4641 q^{76} -2.87564i q^{77} -8.53590 q^{79} +8.73205i q^{82} -8.19615i q^{83} +27.8564 q^{86} +21.4641i q^{88} +5.19615 q^{89} +6.92820 q^{91} -18.9282i q^{92} -14.3923 q^{94} -7.66025i q^{97} +14.7321i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4}+O(q^{10}) \) 4 * q - 8 * q^4	\( 4 q - 8 q^{4} - 16 q^{11} + 32 q^{16} - 4 q^{19} + 32 q^{26} + 8 q^{29} - 12 q^{31} + 8 q^{34} - 8 q^{41} + 8 q^{44} - 24 q^{46} - 20 q^{49} + 48 q^{56} + 40 q^{59} + 16 q^{61} - 64 q^{64} - 8 q^{71} + 8 q^{74} - 40 q^{76} - 48 q^{79} + 56 q^{86} - 16 q^{94}+O(q^{100}) \) 4 * q - 8 * q^4 - 16 * q^11 + 32 * q^16 - 4 * q^19 + 32 * q^26 + 8 * q^29 - 12 * q^31 + 8 * q^34 - 8 * q^41 + 8 * q^44 - 24 * q^46 - 20 * q^49 + 48 * q^56 + 40 * q^59 + 16 * q^61 - 64 * q^64 - 8 * q^71 + 8 * q^74 - 40 * q^76 - 48 * q^79 + 56 * q^86 - 16 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more