LMFDB - Embedded newform 2025.2.b.k.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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 81)
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.k.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} -3.46410 q^{11} +1.00000i q^{13} -3.46410 q^{14} -5.00000 q^{16} +5.19615i q^{17} -2.00000 q^{19} -6.00000i q^{22} -3.46410i q^{23} -1.73205 q^{26} -2.00000i q^{28} -1.73205 q^{29} +8.00000 q^{31} -5.19615i q^{32} -9.00000 q^{34} -7.00000i q^{37} -3.46410i q^{38} -6.92820 q^{41} -2.00000i q^{43} +3.46410 q^{44} +6.00000 q^{46} +6.92820i q^{47} +3.00000 q^{49} -1.00000i q^{52} -3.46410 q^{56} -3.00000i q^{58} -13.8564 q^{59} -7.00000 q^{61} +13.8564i q^{62} -1.00000 q^{64} -10.0000i q^{67} -5.19615i q^{68} -10.3923 q^{71} +7.00000i q^{73} +12.1244 q^{74} +2.00000 q^{76} -6.92820i q^{77} -2.00000 q^{79} -12.0000i q^{82} +13.8564i q^{83} +3.46410 q^{86} -6.00000i q^{88} +5.19615 q^{89} -2.00000 q^{91} +3.46410i q^{92} -12.0000 q^{94} +2.00000i q^{97} +5.19615i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 20 q^{16} - 8 q^{19} + 32 q^{31} - 36 q^{34} + 24 q^{46} + 12 q^{49} - 28 q^{61} - 4 q^{64} + 8 q^{76} - 8 q^{79} - 8 q^{91} - 48 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 - 20 * q^16 - 8 * q^19 + 32 * q^31 - 36 * q^34 + 24 * q^46 + 12 * q^49 - 28 * q^61 - 4 * q^64 + 8 * q^76 - 8 * q^79 - 8 * q^91 - 48 * q^94

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 $

$2$	1.73205i	1.22474i	0.790569	+	0.612372i	$0.209785\pi$
$2$	1.73205i	1.22474i	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	1.73205i	0.612372i
$9$	0	0
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	−3.46410	−0.925820
$15$	0	0
$16$	−5.00000	−1.25000
$17$	5.19615i	1.26025i	0.776493	+	0.630126i	$0.216997\pi$
$17$	5.19615i	1.26025i	−0.776493	+	0.630126i	$0.783003\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	− 6.00000i	− 1.27920i
$23$	− 3.46410i	− 0.722315i	−0.932505	−	0.361158i	$-0.882382\pi$
$23$	− 3.46410i	− 0.722315i	0.932505	−	0.361158i	$-0.117618\pi$
$24$	0	0
$25$	0	0
$26$	−1.73205	−0.339683
$27$	0	0
$28$	− 2.00000i	− 0.377964i
$29$	−1.73205	−0.321634	−0.160817	−	0.986984i	$-0.551413\pi$
$29$	−1.73205	−0.321634	−0.160817	+	0.986984i	$0.551413\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	− 5.19615i	− 0.918559i
$33$	0	0
$34$	−9.00000	−1.54349
$35$	0	0
$36$	0	0
$37$	− 7.00000i	− 1.15079i	−0.817875	−	0.575396i	$-0.804848\pi$
$37$	− 7.00000i	− 1.15079i	0.817875	−	0.575396i	$-0.195152\pi$
$38$	− 3.46410i	− 0.561951i
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	− 2.00000i	− 0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$	− 2.00000i	− 0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$	3.46410	0.522233
$45$	0	0
$46$	6.00000	0.884652
$47$	6.92820i	1.01058i	0.862949	+	0.505291i	$0.168615\pi$
$47$	6.92820i	1.01058i	−0.862949	+	0.505291i	$0.831385\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	−3.46410	−0.462910
$57$	0	0
$58$	− 3.00000i	− 0.393919i
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	0	0
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	13.8564i	1.75977i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	− 5.19615i	− 0.630126i
$69$	0	0
$70$	0	0
$71$	−10.3923	−1.23334	−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923	−1.23334	−0.616670	+	0.787222i	$0.711519\pi$
$72$	0	0
$73$	7.00000i	0.819288i	0.912245	+	0.409644i	$0.134347\pi$
$73$	7.00000i	0.819288i	−0.912245	+	0.409644i	$0.865653\pi$
$74$	12.1244	1.40943
$75$	0	0
$76$	2.00000	0.229416
$77$	− 6.92820i	− 0.789542i
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	0	0
$82$	− 12.0000i	− 1.32518i
$83$	13.8564i	1.52094i	0.649374	+	0.760469i	$0.275031\pi$
$83$	13.8564i	1.52094i	−0.649374	+	0.760469i	$0.724969\pi$
$84$	0	0
$85$	0	0
$86$	3.46410	0.373544
$87$	0	0
$88$	− 6.00000i	− 0.639602i
$89$	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
$91$	−2.00000	−0.209657
$92$	3.46410i	0.361158i
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	5.19615i	0.524891i
$99$	0	0
$100$	0	0
$101$	6.92820	0.689382	0.344691	−	0.938716i	$-0.387984\pi$
$101$	6.92820	0.689382	0.344691	+	0.938716i	$0.387984\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−1.73205	−0.169842
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−11.0000	−1.05361	−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000	−1.05361	−0.526804	+	0.849987i	$0.676610\pi$
$110$	0	0
$111$	0	0
$112$	− 10.0000i	− 0.944911i
$113$	1.73205i	0.162938i	0.996676	+	0.0814688i	$0.0259611\pi$
$113$	1.73205i	0.162938i	−0.996676	+	0.0814688i	$0.974039\pi$
$114$	0	0
$115$	0	0
$116$	1.73205	0.160817
$117$	0	0
$118$	− 24.0000i	− 2.20938i
$119$	−10.3923	−0.952661
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 12.1244i	− 1.09769i
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	2.00000i	0.177471i	0.996055	+	0.0887357i	$0.0282826\pi$
$127$	2.00000i	0.177471i	−0.996055	+	0.0887357i	$0.971717\pi$
$128$	− 12.1244i	− 1.07165i
$129$	0	0
$130$	0	0
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	17.3205	1.49626
$135$	0	0
$136$	−9.00000	−0.771744
$137$	1.73205i	0.147979i	0.997259	+	0.0739895i	$0.0235731\pi$
$137$	1.73205i	0.147979i	−0.997259	+	0.0739895i	$0.976427\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	− 18.0000i	− 1.51053i
$143$	− 3.46410i	− 0.289683i
$144$	0	0
$145$	0	0
$146$	−12.1244	−1.00342
$147$	0	0
$148$	7.00000i	0.575396i
$149$	−8.66025	−0.709476	−0.354738	−	0.934966i	$-0.615430\pi$
$149$	−8.66025	−0.709476	−0.354738	+	0.934966i	$0.615430\pi$
$150$	0	0
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	− 3.46410i	− 0.280976i
$153$	0	0
$154$	12.0000	0.966988
$155$	0	0
$156$	0	0
$157$	17.0000i	1.35675i	0.734717	+	0.678374i	$0.237315\pi$
$157$	17.0000i	1.35675i	−0.734717	+	0.678374i	$0.762685\pi$
$158$	− 3.46410i	− 0.275589i
$159$	0	0
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	6.92820	0.541002
$165$	0	0
$166$	−24.0000	−1.86276
$167$	− 17.3205i	− 1.34030i	−0.742225	−	0.670151i	$-0.766230\pi$
$167$	− 17.3205i	− 1.34030i	0.742225	−	0.670151i	$-0.233770\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	2.00000i	0.152499i
$173$	19.0526i	1.44854i	0.689517	+	0.724270i	$0.257823\pi$
$173$	19.0526i	1.44854i	−0.689517	+	0.724270i	$0.742177\pi$
$174$	0	0
$175$	0	0
$176$	17.3205	1.30558
$177$	0	0
$178$	9.00000i	0.674579i
$179$	−20.7846	−1.55351	−0.776757	−	0.629800i	$-0.783137\pi$
$179$	−20.7846	−1.55351	−0.776757	+	0.629800i	$0.783137\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	− 3.46410i	− 0.256776i
$183$	0	0
$184$	6.00000	0.442326
$185$	0	0
$186$	0	0
$187$	− 18.0000i	− 1.31629i
$188$	− 6.92820i	− 0.505291i
$189$	0	0
$190$	0	0
$191$	17.3205	1.25327	0.626634	−	0.779314i	$-0.284432\pi$
$191$	17.3205	1.25327	0.626634	+	0.779314i	$0.284432\pi$
$192$	0	0
$193$	1.00000i	0.0719816i	0.999352	+	0.0359908i	$0.0114587\pi$
$193$	1.00000i	0.0719816i	−0.999352	+	0.0359908i	$0.988541\pi$
$194$	−3.46410	−0.248708
$195$	0	0
$196$	−3.00000	−0.214286
$197$	− 5.19615i	− 0.370211i	−0.982719	−	0.185105i	$-0.940737\pi$
$197$	− 5.19615i	− 0.370211i	0.982719	−	0.185105i	$-0.0592626\pi$
$198$	0	0
$199$	−20.0000	−1.41776	−0.708881	−	0.705328i	$-0.750800\pi$
$199$	−20.0000	−1.41776	−0.708881	+	0.705328i	$0.750800\pi$
$200$	0	0
$201$	0	0
$202$	12.0000i	0.844317i
$203$	− 3.46410i	− 0.243132i
$204$	0	0
$205$	0	0
$206$	13.8564	0.965422
$207$	0	0
$208$	− 5.00000i	− 0.346688i
$209$	6.92820	0.479234
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	− 19.0526i	− 1.29040i
$219$	0	0
$220$	0	0
$221$	−5.19615	−0.349531
$222$	0	0
$223$	− 2.00000i	− 0.133930i	−0.997755	−	0.0669650i	$-0.978668\pi$
$223$	− 2.00000i	− 0.133930i	0.997755	−	0.0669650i	$-0.0213316\pi$
$224$	10.3923	0.694365
$225$	0	0
$226$	−3.00000	−0.199557
$227$	− 3.46410i	− 0.229920i	−0.993370	−	0.114960i	$-0.963326\pi$
$227$	− 3.46410i	− 0.229920i	0.993370	−	0.114960i	$-0.0366741\pi$
$228$	0	0
$229$	1.00000	0.0660819	0.0330409	−	0.999454i	$-0.489481\pi$
$229$	1.00000	0.0660819	0.0330409	+	0.999454i	$0.489481\pi$
$230$	0	0
$231$	0	0
$232$	− 3.00000i	− 0.196960i
$233$	25.9808i	1.70206i	0.525120	+	0.851028i	$0.324020\pi$
$233$	25.9808i	1.70206i	−0.525120	+	0.851028i	$0.675980\pi$
$234$	0	0
$235$	0	0
$236$	13.8564	0.901975
$237$	0	0
$238$	− 18.0000i	− 1.16677i
$239$	27.7128	1.79259	0.896296	−	0.443455i	$-0.146248\pi$
$239$	27.7128	1.79259	0.896296	+	0.443455i	$0.146248\pi$
$240$	0	0
$241$	29.0000	1.86805	0.934027	−	0.357202i	$-0.116269\pi$
$241$	29.0000	1.86805	0.934027	+	0.357202i	$0.116269\pi$
$242$	1.73205i	0.111340i
$243$	0	0
$244$	7.00000	0.448129
$245$	0	0
$246$	0	0
$247$	− 2.00000i	− 0.127257i
$248$	13.8564i	0.879883i
$249$	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
$253$	12.0000i	0.754434i
$254$	−3.46410	−0.217357
$255$	0	0
$256$	19.0000	1.18750
$257$	8.66025i	0.540212i	0.962831	+	0.270106i	$0.0870587\pi$
$257$	8.66025i	0.540212i	−0.962831	+	0.270106i	$0.912941\pi$
$258$	0	0
$259$	14.0000	0.869918
$260$	0	0
$261$	0	0
$262$	6.00000i	0.370681i
$263$	− 6.92820i	− 0.427211i	−0.976920	−	0.213606i	$-0.931479\pi$
$263$	− 6.92820i	− 0.427211i	0.976920	−	0.213606i	$-0.0685208\pi$
$264$	0	0
$265$	0	0
$266$	6.92820	0.424795
$267$	0	0
$268$	10.0000i	0.610847i
$269$	15.5885	0.950445	0.475223	−	0.879866i	$-0.342368\pi$
$269$	15.5885	0.950445	0.475223	+	0.879866i	$0.342368\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	− 25.9808i	− 1.57532i
$273$	0	0
$274$	−3.00000	−0.181237
$275$	0	0
$276$	0	0
$277$	2.00000i	0.120168i	0.998193	+	0.0600842i	$0.0191369\pi$
$277$	2.00000i	0.120168i	−0.998193	+	0.0600842i	$0.980863\pi$
$278$	− 13.8564i	− 0.831052i
$279$	0	0
$280$	0	0
$281$	12.1244	0.723278	0.361639	−	0.932318i	$-0.382217\pi$
$281$	12.1244	0.723278	0.361639	+	0.932318i	$0.382217\pi$
$282$	0	0
$283$	28.0000i	1.66443i	0.554455	+	0.832214i	$0.312927\pi$
$283$	28.0000i	1.66443i	−0.554455	+	0.832214i	$0.687073\pi$
$284$	10.3923	0.616670
$285$	0	0
$286$	6.00000	0.354787
$287$	− 13.8564i	− 0.817918i
$288$	0	0
$289$	−10.0000	−0.588235
$290$	0	0
$291$	0	0
$292$	− 7.00000i	− 0.409644i
$293$	− 19.0526i	− 1.11306i	−0.830827	−	0.556531i	$-0.812132\pi$
$293$	− 19.0526i	− 1.11306i	0.830827	−	0.556531i	$-0.187868\pi$
$294$	0	0
$295$	0	0
$296$	12.1244	0.704714
$297$	0	0
$298$	− 15.0000i	− 0.868927i
$299$	3.46410	0.200334
$300$	0	0
$301$	4.00000	0.230556
$302$	34.6410i	1.99337i
$303$	0	0
$304$	10.0000	0.573539
$305$	0	0
$306$	0	0
$307$	− 16.0000i	− 0.913168i	−0.889680	−	0.456584i	$-0.849073\pi$
$307$	− 16.0000i	− 0.913168i	0.889680	−	0.456584i	$-0.150927\pi$
$308$	6.92820i	0.394771i
$309$	0	0
$310$	0	0
$311$	−6.92820	−0.392862	−0.196431	−	0.980518i	$-0.562935\pi$
$311$	−6.92820	−0.392862	−0.196431	+	0.980518i	$0.562935\pi$
$312$	0	0
$313$	25.0000i	1.41308i	0.707671	+	0.706542i	$0.249746\pi$
$313$	25.0000i	1.41308i	−0.707671	+	0.706542i	$0.750254\pi$
$314$	−29.4449	−1.66167
$315$	0	0
$316$	2.00000	0.112509
$317$	− 8.66025i	− 0.486408i	−0.969975	−	0.243204i	$-0.921801\pi$
$317$	− 8.66025i	− 0.486408i	0.969975	−	0.243204i	$-0.0781985\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	12.0000i	0.668734i
$323$	− 10.3923i	− 0.578243i
$324$	0	0
$325$	0	0
$326$	−27.7128	−1.53487
$327$	0	0
$328$	− 12.0000i	− 0.662589i
$329$	−13.8564	−0.763928
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	− 13.8564i	− 0.760469i
$333$	0	0
$334$	30.0000	1.64153
$335$	0	0
$336$	0	0
$337$	26.0000i	1.41631i	0.706057	+	0.708155i	$0.250472\pi$
$337$	26.0000i	1.41631i	−0.706057	+	0.708155i	$0.749528\pi$
$338$	20.7846i	1.13053i
$339$	0	0
$340$	0	0
$341$	−27.7128	−1.50073
$342$	0	0
$343$	20.0000i	1.07990i
$344$	3.46410	0.186772
$345$	0	0
$346$	−33.0000	−1.77409
$347$	3.46410i	0.185963i	0.995668	+	0.0929814i	$0.0296397\pi$
$347$	3.46410i	0.185963i	−0.995668	+	0.0929814i	$0.970360\pi$
$348$	0	0
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	0	0
$352$	18.0000i	0.959403i
$353$	13.8564i	0.737502i	0.929528	+	0.368751i	$0.120215\pi$
$353$	13.8564i	0.737502i	−0.929528	+	0.368751i	$0.879785\pi$
$354$	0	0
$355$	0	0
$356$	−5.19615	−0.275396
$357$	0	0
$358$	− 36.0000i	− 1.90266i
$359$	−10.3923	−0.548485	−0.274242	−	0.961661i	$-0.588427\pi$
$359$	−10.3923	−0.548485	−0.274242	+	0.961661i	$0.588427\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	3.46410i	0.182069i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	20.0000i	1.04399i	0.852948	+	0.521996i	$0.174812\pi$
$367$	20.0000i	1.04399i	−0.852948	+	0.521996i	$0.825188\pi$
$368$	17.3205i	0.902894i
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	31.1769	1.61212
$375$	0	0
$376$	−12.0000	−0.618853
$377$	− 1.73205i	− 0.0892052i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	0	0
$382$	30.0000i	1.53493i
$383$	17.3205i	0.885037i	0.896759	+	0.442518i	$0.145915\pi$
$383$	17.3205i	0.885037i	−0.896759	+	0.442518i	$0.854085\pi$
$384$	0	0
$385$	0	0
$386$	−1.73205	−0.0881591
$387$	0	0
$388$	− 2.00000i	− 0.101535i
$389$	−27.7128	−1.40510	−0.702548	−	0.711637i	$-0.747954\pi$
$389$	−27.7128	−1.40510	−0.702548	+	0.711637i	$0.747954\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	5.19615i	0.262445i
$393$	0	0
$394$	9.00000	0.453413
$395$	0	0
$396$	0	0
$397$	29.0000i	1.45547i	0.685859	+	0.727734i	$0.259427\pi$
$397$	29.0000i	1.45547i	−0.685859	+	0.727734i	$0.740573\pi$
$398$	− 34.6410i	− 1.73640i
$399$	0	0
$400$	0	0
$401$	−12.1244	−0.605461	−0.302731	−	0.953076i	$-0.597898\pi$
$401$	−12.1244	−0.605461	−0.302731	+	0.953076i	$0.597898\pi$
$402$	0	0
$403$	8.00000i	0.398508i
$404$	−6.92820	−0.344691
$405$	0	0
$406$	6.00000	0.297775
$407$	24.2487i	1.20196i
$408$	0	0
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	0	0
$412$	8.00000i	0.394132i
$413$	− 27.7128i	− 1.36366i
$414$	0	0
$415$	0	0
$416$	5.19615	0.254762
$417$	0	0
$418$	12.0000i	0.586939i
$419$	6.92820	0.338465	0.169232	−	0.985576i	$-0.445871\pi$
$419$	6.92820	0.338465	0.169232	+	0.985576i	$0.445871\pi$
$420$	0	0
$421$	−25.0000	−1.21843	−0.609213	−	0.793007i	$-0.708514\pi$
$421$	−25.0000	−1.21843	−0.609213	+	0.793007i	$0.708514\pi$
$422$	− 17.3205i	− 0.843149i
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 14.0000i	− 0.677507i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	− 11.0000i	− 0.528626i	−0.964437	−	0.264313i	$-0.914855\pi$
$433$	− 11.0000i	− 0.528626i	0.964437	−	0.264313i	$-0.0851452\pi$
$434$	−27.7128	−1.33026
$435$	0	0
$436$	11.0000	0.526804
$437$	6.92820i	0.331421i
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	0	0
$442$	− 9.00000i	− 0.428086i
$443$	34.6410i	1.64584i	0.568154	+	0.822922i	$0.307658\pi$
$443$	34.6410i	1.64584i	−0.568154	+	0.822922i	$0.692342\pi$
$444$	0	0
$445$	0	0
$446$	3.46410	0.164030
$447$	0	0
$448$	− 2.00000i	− 0.0944911i
$449$	−20.7846	−0.980886	−0.490443	−	0.871473i	$-0.663165\pi$
$449$	−20.7846	−0.980886	−0.490443	+	0.871473i	$0.663165\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	− 1.73205i	− 0.0814688i
$453$	0	0
$454$	6.00000	0.281594
$455$	0	0
$456$	0	0
$457$	29.0000i	1.35656i	0.734802	+	0.678281i	$0.237275\pi$
$457$	29.0000i	1.35656i	−0.734802	+	0.678281i	$0.762725\pi$
$458$	1.73205i	0.0809334i
$459$	0	0
$460$	0	0
$461$	−13.8564	−0.645357	−0.322679	−	0.946509i	$-0.604583\pi$
$461$	−13.8564	−0.645357	−0.322679	+	0.946509i	$0.604583\pi$
$462$	0	0
$463$	− 8.00000i	− 0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$	− 8.00000i	− 0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$	8.66025	0.402042
$465$	0	0
$466$	−45.0000	−2.08458
$467$	− 20.7846i	− 0.961797i	−0.876776	−	0.480899i	$-0.840311\pi$
$467$	− 20.7846i	− 0.961797i	0.876776	−	0.480899i	$-0.159689\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	0	0
$472$	− 24.0000i	− 1.10469i
$473$	6.92820i	0.318559i
$474$	0	0
$475$	0	0
$476$	10.3923	0.476331
$477$	0	0
$478$	48.0000i	2.19547i
$479$	24.2487	1.10795	0.553976	−	0.832533i	$-0.313110\pi$
$479$	24.2487	1.10795	0.553976	+	0.832533i	$0.313110\pi$
$480$	0	0
$481$	7.00000	0.319173
$482$	50.2295i	2.28789i
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	0	0
$487$	− 16.0000i	− 0.725029i	−0.931978	−	0.362515i	$-0.881918\pi$
$487$	− 16.0000i	− 0.725029i	0.931978	−	0.362515i	$-0.118082\pi$
$488$	− 12.1244i	− 0.548844i
$489$	0	0
$490$	0	0
$491$	−17.3205	−0.781664	−0.390832	−	0.920462i	$-0.627813\pi$
$491$	−17.3205	−0.781664	−0.390832	+	0.920462i	$0.627813\pi$
$492$	0	0
$493$	− 9.00000i	− 0.405340i
$494$	3.46410	0.155857
$495$	0	0
$496$	−40.0000	−1.79605
$497$	− 20.7846i	− 0.932317i
$498$	0	0
$499$	10.0000	0.447661	0.223831	−	0.974628i	$-0.428144\pi$
$499$	10.0000	0.447661	0.223831	+	0.974628i	$0.428144\pi$
$500$	0	0
$501$	0	0
$502$	18.0000i	0.803379i
$503$	− 20.7846i	− 0.926740i	−0.886165	−	0.463370i	$-0.846640\pi$
$503$	− 20.7846i	− 0.926740i	0.886165	−	0.463370i	$-0.153360\pi$
$504$	0	0
$505$	0	0
$506$	−20.7846	−0.923989
$507$	0	0
$508$	− 2.00000i	− 0.0887357i
$509$	27.7128	1.22835	0.614174	−	0.789170i	$-0.289489\pi$
$509$	27.7128	1.22835	0.614174	+	0.789170i	$0.289489\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	8.66025i	0.382733i
$513$	0	0
$514$	−15.0000	−0.661622
$515$	0	0
$516$	0	0
$517$	− 24.0000i	− 1.05552i
$518$	24.2487i	1.06543i
$519$	0	0
$520$	0	0
$521$	−20.7846	−0.910590	−0.455295	−	0.890341i	$-0.650466\pi$
$521$	−20.7846	−0.910590	−0.455295	+	0.890341i	$0.650466\pi$
$522$	0	0
$523$	− 38.0000i	− 1.66162i	−0.556553	−	0.830812i	$-0.687876\pi$
$523$	− 38.0000i	− 1.66162i	0.556553	−	0.830812i	$-0.312124\pi$
$524$	−3.46410	−0.151330
$525$	0	0
$526$	12.0000	0.523225
$527$	41.5692i	1.81078i
$528$	0	0
$529$	11.0000	0.478261
$530$	0	0
$531$	0	0
$532$	4.00000i	0.173422i
$533$	− 6.92820i	− 0.300094i
$534$	0	0
$535$	0	0
$536$	17.3205	0.748132
$537$	0	0
$538$	27.0000i	1.16405i
$539$	−10.3923	−0.447628
$540$	0	0
$541$	11.0000	0.472927	0.236463	−	0.971640i	$-0.424012\pi$
$541$	11.0000	0.472927	0.236463	+	0.971640i	$0.424012\pi$
$542$	3.46410i	0.148796i
$543$	0	0
$544$	27.0000	1.15762
$545$	0	0
$546$	0	0
$547$	20.0000i	0.855138i	0.903983	+	0.427569i	$0.140630\pi$
$547$	20.0000i	0.855138i	−0.903983	+	0.427569i	$0.859370\pi$
$548$	− 1.73205i	− 0.0739895i
$549$	0	0
$550$	0	0
$551$	3.46410	0.147576
$552$	0	0
$553$	− 4.00000i	− 0.170097i
$554$	−3.46410	−0.147176
$555$	0	0
$556$	8.00000	0.339276
$557$	36.3731i	1.54118i	0.637333	+	0.770588i	$0.280037\pi$
$557$	36.3731i	1.54118i	−0.637333	+	0.770588i	$0.719963\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	0	0
$562$	21.0000i	0.885832i
$563$	− 34.6410i	− 1.45994i	−0.683477	−	0.729972i	$-0.739533\pi$
$563$	− 34.6410i	− 1.45994i	0.683477	−	0.729972i	$-0.260467\pi$
$564$	0	0
$565$	0	0
$566$	−48.4974	−2.03850
$567$	0	0
$568$	− 18.0000i	− 0.755263i
$569$	−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
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	3.46410i	0.144841i
$573$	0	0
$574$	24.0000	1.00174
$575$	0	0
$576$	0	0
$577$	11.0000i	0.457936i	0.973434	+	0.228968i	$0.0735351\pi$
$577$	11.0000i	0.457936i	−0.973434	+	0.228968i	$0.926465\pi$
$578$	− 17.3205i	− 0.720438i
$579$	0	0
$580$	0	0
$581$	−27.7128	−1.14972
$582$	0	0
$583$	0	0
$584$	−12.1244	−0.501709
$585$	0	0
$586$	33.0000	1.36322
$587$	38.1051i	1.57277i	0.617739	+	0.786383i	$0.288049\pi$
$587$	38.1051i	1.57277i	−0.617739	+	0.786383i	$0.711951\pi$
$588$	0	0
$589$	−16.0000	−0.659269
$590$	0	0
$591$	0	0
$592$	35.0000i	1.43849i
$593$	− 15.5885i	− 0.640141i	−0.947394	−	0.320071i	$-0.896293\pi$
$593$	− 15.5885i	− 0.640141i	0.947394	−	0.320071i	$-0.103707\pi$
$594$	0	0
$595$	0	0
$596$	8.66025	0.354738
$597$	0	0
$598$	6.00000i	0.245358i
$599$	−13.8564	−0.566157	−0.283079	−	0.959097i	$-0.591356\pi$
$599$	−13.8564	−0.566157	−0.283079	+	0.959097i	$0.591356\pi$
$600$	0	0
$601$	−25.0000	−1.01977	−0.509886	−	0.860242i	$-0.670312\pi$
$601$	−25.0000	−1.01977	−0.509886	+	0.860242i	$0.670312\pi$
$602$	6.92820i	0.282372i
$603$	0	0
$604$	−20.0000	−0.813788
$605$	0	0
$606$	0	0
$607$	26.0000i	1.05531i	0.849460	+	0.527654i	$0.176928\pi$
$607$	26.0000i	1.05531i	−0.849460	+	0.527654i	$0.823072\pi$
$608$	10.3923i	0.421464i
$609$	0	0
$610$	0	0
$611$	−6.92820	−0.280285
$612$	0	0
$613$	34.0000i	1.37325i	0.727013	+	0.686624i	$0.240908\pi$
$613$	34.0000i	1.37325i	−0.727013	+	0.686624i	$0.759092\pi$
$614$	27.7128	1.11840
$615$	0	0
$616$	12.0000	0.483494
$617$	− 12.1244i	− 0.488108i	−0.969762	−	0.244054i	$-0.921523\pi$
$617$	− 12.1244i	− 0.488108i	0.969762	−	0.244054i	$-0.0784774\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	− 12.0000i	− 0.481156i
$623$	10.3923i	0.416359i
$624$	0	0
$625$	0	0
$626$	−43.3013	−1.73067
$627$	0	0
$628$	− 17.0000i	− 0.678374i
$629$	36.3731	1.45029
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	− 3.46410i	− 0.137795i
$633$	0	0
$634$	15.0000	0.595726
$635$	0	0
$636$	0	0
$637$	3.00000i	0.118864i
$638$	10.3923i	0.411435i
$639$	0	0
$640$	0	0
$641$	22.5167	0.889355	0.444677	−	0.895691i	$-0.353318\pi$
$641$	22.5167	0.889355	0.444677	+	0.895691i	$0.353318\pi$
$642$	0	0
$643$	− 8.00000i	− 0.315489i	−0.987480	−	0.157745i	$-0.949578\pi$
$643$	− 8.00000i	− 0.315489i	0.987480	−	0.157745i	$-0.0504223\pi$
$644$	−6.92820	−0.273009
$645$	0	0
$646$	18.0000	0.708201
$647$	− 31.1769i	− 1.22569i	−0.790203	−	0.612845i	$-0.790025\pi$
$647$	− 31.1769i	− 1.22569i	0.790203	−	0.612845i	$-0.209975\pi$
$648$	0	0
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	− 16.0000i	− 0.626608i
$653$	− 13.8564i	− 0.542243i	−0.962545	−	0.271122i	$-0.912605\pi$
$653$	− 13.8564i	− 0.542243i	0.962545	−	0.271122i	$-0.0873945\pi$
$654$	0	0
$655$	0	0
$656$	34.6410	1.35250
$657$	0	0
$658$	− 24.0000i	− 0.935617i
$659$	3.46410	0.134942	0.0674711	−	0.997721i	$-0.478507\pi$
$659$	3.46410	0.134942	0.0674711	+	0.997721i	$0.478507\pi$
$660$	0	0
$661$	17.0000	0.661223	0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000	0.661223	0.330612	+	0.943767i	$0.392745\pi$
$662$	3.46410i	0.134636i
$663$	0	0
$664$	−24.0000	−0.931381
$665$	0	0
$666$	0	0
$667$	6.00000i	0.232321i
$668$	17.3205i	0.670151i
$669$	0	0
$670$	0	0
$671$	24.2487	0.936111
$672$	0	0
$673$	25.0000i	0.963679i	0.876259	+	0.481840i	$0.160031\pi$
$673$	25.0000i	0.963679i	−0.876259	+	0.481840i	$0.839969\pi$
$674$	−45.0333	−1.73462
$675$	0	0
$676$	−12.0000	−0.461538
$677$	− 13.8564i	− 0.532545i	−0.963898	−	0.266272i	$-0.914208\pi$
$677$	− 13.8564i	− 0.532545i	0.963898	−	0.266272i	$-0.0857921\pi$
$678$	0	0
$679$	−4.00000	−0.153506
$680$	0	0
$681$	0	0
$682$	− 48.0000i	− 1.83801i
$683$	20.7846i	0.795301i	0.917537	+	0.397650i	$0.130174\pi$
$683$	20.7846i	0.795301i	−0.917537	+	0.397650i	$0.869826\pi$
$684$	0	0
$685$	0	0
$686$	−34.6410	−1.32260
$687$	0	0
$688$	10.0000i	0.381246i
$689$	0	0
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	− 19.0526i	− 0.724270i
$693$	0	0
$694$	−6.00000	−0.227757
$695$	0	0
$696$	0	0
$697$	− 36.0000i	− 1.36360i
$698$	− 3.46410i	− 0.131118i
$699$	0	0
$700$	0	0
$701$	46.7654	1.76630	0.883152	−	0.469087i	$-0.155417\pi$
$701$	46.7654	1.76630	0.883152	+	0.469087i	$0.155417\pi$
$702$	0	0
$703$	14.0000i	0.528020i
$704$	3.46410	0.130558
$705$	0	0
$706$	−24.0000	−0.903252
$707$	13.8564i	0.521124i
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	0	0
$712$	9.00000i	0.337289i
$713$	− 27.7128i	− 1.03785i
$714$	0	0
$715$	0	0
$716$	20.7846	0.776757
$717$	0	0
$718$	− 18.0000i	− 0.671754i
$719$	10.3923	0.387568	0.193784	−	0.981044i	$-0.437924\pi$
$719$	10.3923	0.387568	0.193784	+	0.981044i	$0.437924\pi$
$720$	0	0
$721$	16.0000	0.595871
$722$	− 25.9808i	− 0.966904i
$723$	0	0
$724$	−2.00000	−0.0743294
$725$	0	0
$726$	0	0
$727$	− 34.0000i	− 1.26099i	−0.776193	−	0.630495i	$-0.782852\pi$
$727$	− 34.0000i	− 1.26099i	0.776193	−	0.630495i	$-0.217148\pi$
$728$	− 3.46410i	− 0.128388i
$729$	0	0
$730$	0	0
$731$	10.3923	0.384373
$732$	0	0
$733$	46.0000i	1.69905i	0.527549	+	0.849524i	$0.323111\pi$
$733$	46.0000i	1.69905i	−0.527549	+	0.849524i	$0.676889\pi$
$734$	−34.6410	−1.27862
$735$	0	0
$736$	−18.0000	−0.663489
$737$	34.6410i	1.27602i
$738$	0	0
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.92820i	0.254171i	0.991892	+	0.127086i	$0.0405623\pi$
$743$	6.92820i	0.254171i	−0.991892	+	0.127086i	$0.959438\pi$
$744$	0	0
$745$	0	0
$746$	−17.3205	−0.634149
$747$	0	0
$748$	18.0000i	0.658145i
$749$	0	0
$750$	0	0
$751$	−10.0000	−0.364905	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	−10.0000	−0.364905	−0.182453	+	0.983215i	$0.558404\pi$
$752$	− 34.6410i	− 1.26323i
$753$	0	0
$754$	3.00000	0.109254
$755$	0	0
$756$	0	0
$757$	2.00000i	0.0726912i	0.999339	+	0.0363456i	$0.0115717\pi$
$757$	2.00000i	0.0726912i	−0.999339	+	0.0363456i	$0.988428\pi$
$758$	27.7128i	1.00657i
$759$	0	0
$760$	0	0
$761$	29.4449	1.06738	0.533688	−	0.845682i	$-0.320806\pi$
$761$	29.4449	1.06738	0.533688	+	0.845682i	$0.320806\pi$
$762$	0	0
$763$	− 22.0000i	− 0.796453i
$764$	−17.3205	−0.626634
$765$	0	0
$766$	−30.0000	−1.08394
$767$	− 13.8564i	− 0.500326i
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	0	0
$772$	− 1.00000i	− 0.0359908i
$773$	25.9808i	0.934463i	0.884135	+	0.467232i	$0.154749\pi$
$773$	25.9808i	0.934463i	−0.884135	+	0.467232i	$0.845251\pi$
$774$	0	0
$775$	0	0
$776$	−3.46410	−0.124354
$777$	0	0
$778$	− 48.0000i	− 1.72088i
$779$	13.8564	0.496457
$780$	0	0
$781$	36.0000	1.28818
$782$	31.1769i	1.11488i
$783$	0	0
$784$	−15.0000	−0.535714
$785$	0	0
$786$	0	0
$787$	26.0000i	0.926800i	0.886149	+	0.463400i	$0.153371\pi$
$787$	26.0000i	0.926800i	−0.886149	+	0.463400i	$0.846629\pi$
$788$	5.19615i	0.185105i
$789$	0	0
$790$	0	0
$791$	−3.46410	−0.123169
$792$	0	0
$793$	− 7.00000i	− 0.248577i
$794$	−50.2295	−1.78258
$795$	0	0
$796$	20.0000	0.708881
$797$	− 53.6936i	− 1.90192i	−0.309308	−	0.950962i	$-0.600097\pi$
$797$	− 53.6936i	− 1.90192i	0.309308	−	0.950962i	$-0.399903\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	0	0
$802$	− 21.0000i	− 0.741536i
$803$	− 24.2487i	− 0.855718i
$804$	0	0
$805$	0	0
$806$	−13.8564	−0.488071
$807$	0	0
$808$	12.0000i	0.422159i
$809$	46.7654	1.64418	0.822091	−	0.569355i	$-0.192807\pi$
$809$	46.7654	1.64418	0.822091	+	0.569355i	$0.192807\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	3.46410i	0.121566i
$813$	0	0
$814$	−42.0000	−1.47210
$815$	0	0
$816$	0	0
$817$	4.00000i	0.139942i
$818$	32.9090i	1.15063i
$819$	0	0
$820$	0	0
$821$	12.1244	0.423143	0.211571	−	0.977363i	$-0.432142\pi$
$821$	12.1244	0.423143	0.211571	+	0.977363i	$0.432142\pi$
$822$	0	0
$823$	28.0000i	0.976019i	0.872838	+	0.488009i	$0.162277\pi$
$823$	28.0000i	0.976019i	−0.872838	+	0.488009i	$0.837723\pi$
$824$	13.8564	0.482711
$825$	0	0
$826$	48.0000	1.67013
$827$	− 10.3923i	− 0.361376i	−0.983540	−	0.180688i	$-0.942168\pi$
$827$	− 10.3923i	− 0.361376i	0.983540	−	0.180688i	$-0.0578324\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	0	0
$832$	− 1.00000i	− 0.0346688i
$833$	15.5885i	0.540108i
$834$	0	0
$835$	0	0
$836$	−6.92820	−0.239617
$837$	0	0
$838$	12.0000i	0.414533i
$839$	45.0333	1.55472	0.777361	−	0.629054i	$-0.216558\pi$
$839$	45.0333	1.55472	0.777361	+	0.629054i	$0.216558\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	− 43.3013i	− 1.49226i
$843$	0	0
$844$	10.0000	0.344214
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−24.2487	−0.831235
$852$	0	0
$853$	34.0000i	1.16414i	0.813139	+	0.582069i	$0.197757\pi$
$853$	34.0000i	1.16414i	−0.813139	+	0.582069i	$0.802243\pi$
$854$	24.2487	0.829774
$855$	0	0
$856$	0	0
$857$	22.5167i	0.769154i	0.923093	+	0.384577i	$0.125653\pi$
$857$	22.5167i	0.769154i	−0.923093	+	0.384577i	$0.874347\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 31.1769i	− 1.06127i	−0.847599	−	0.530637i	$-0.821953\pi$
$863$	− 31.1769i	− 1.06127i	0.847599	−	0.530637i	$-0.178047\pi$
$864$	0	0
$865$	0	0
$866$	19.0526	0.647432
$867$	0	0
$868$	− 16.0000i	− 0.543075i
$869$	6.92820	0.235023
$870$	0	0
$871$	10.0000	0.338837
$872$	− 19.0526i	− 0.645201i
$873$	0	0
$874$	−12.0000	−0.405906
$875$	0	0
$876$	0	0
$877$	53.0000i	1.78968i	0.446384	+	0.894841i	$0.352711\pi$
$877$	53.0000i	1.78968i	−0.446384	+	0.894841i	$0.647289\pi$
$878$	− 34.6410i	− 1.16908i
$879$	0	0
$880$	0	0
$881$	20.7846	0.700251	0.350126	−	0.936703i	$-0.386139\pi$
$881$	20.7846	0.700251	0.350126	+	0.936703i	$0.386139\pi$
$882$	0	0
$883$	− 56.0000i	− 1.88455i	−0.334840	−	0.942275i	$-0.608682\pi$
$883$	− 56.0000i	− 1.88455i	0.334840	−	0.942275i	$-0.391318\pi$
$884$	5.19615	0.174766
$885$	0	0
$886$	−60.0000	−2.01574
$887$	3.46410i	0.116313i	0.998307	+	0.0581566i	$0.0185223\pi$
$887$	3.46410i	0.116313i	−0.998307	+	0.0581566i	$0.981478\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	2.00000i	0.0669650i
$893$	− 13.8564i	− 0.463687i
$894$	0	0
$895$	0	0
$896$	24.2487	0.810093
$897$	0	0
$898$	− 36.0000i	− 1.20134i
$899$	−13.8564	−0.462137
$900$	0	0
$901$	0	0
$902$	41.5692i	1.38410i
$903$	0	0
$904$	−3.00000	−0.0997785
$905$	0	0
$906$	0	0
$907$	− 52.0000i	− 1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$	− 52.0000i	− 1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$	3.46410i	0.114960i
$909$	0	0
$910$	0	0
$911$	−24.2487	−0.803396	−0.401698	−	0.915772i	$-0.631580\pi$
$911$	−24.2487	−0.803396	−0.401698	+	0.915772i	$0.631580\pi$
$912$	0	0
$913$	− 48.0000i	− 1.58857i
$914$	−50.2295	−1.66144
$915$	0	0
$916$	−1.00000	−0.0330409
$917$	6.92820i	0.228789i
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	0	0
$921$	0	0
$922$	− 24.0000i	− 0.790398i
$923$	− 10.3923i	− 0.342067i
$924$	0	0
$925$	0	0
$926$	13.8564	0.455350
$927$	0	0
$928$	9.00000i	0.295439i
$929$	50.2295	1.64798	0.823988	−	0.566608i	$-0.191744\pi$
$929$	50.2295	1.64798	0.823988	+	0.566608i	$0.191744\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	− 25.9808i	− 0.851028i
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	− 25.0000i	− 0.816714i	−0.912822	−	0.408357i	$-0.866102\pi$
$937$	− 25.0000i	− 0.816714i	0.912822	−	0.408357i	$-0.133898\pi$
$938$	34.6410i	1.13107i
$939$	0	0
$940$	0	0
$941$	50.2295	1.63743	0.818717	−	0.574197i	$-0.194686\pi$
$941$	50.2295	1.63743	0.818717	+	0.574197i	$0.194686\pi$
$942$	0	0
$943$	24.0000i	0.781548i
$944$	69.2820	2.25494
$945$	0	0
$946$	−12.0000	−0.390154
$947$	17.3205i	0.562841i	0.959585	+	0.281420i	$0.0908056\pi$
$947$	17.3205i	0.562841i	−0.959585	+	0.281420i	$0.909194\pi$
$948$	0	0
$949$	−7.00000	−0.227230
$950$	0	0
$951$	0	0
$952$	− 18.0000i	− 0.583383i
$953$	5.19615i	0.168320i	0.996452	+	0.0841599i	$0.0268207\pi$
$953$	5.19615i	0.168320i	−0.996452	+	0.0841599i	$0.973179\pi$
$954$	0	0
$955$	0	0
$956$	−27.7128	−0.896296
$957$	0	0
$958$	42.0000i	1.35696i
$959$	−3.46410	−0.111862
$960$	0	0
$961$	33.0000	1.06452
$962$	12.1244i	0.390905i
$963$	0	0
$964$	−29.0000	−0.934027
$965$	0	0
$966$	0	0
$967$	− 46.0000i	− 1.47926i	−0.673014	−	0.739630i	$-0.735000\pi$
$967$	− 46.0000i	− 1.47926i	0.673014	−	0.739630i	$-0.265000\pi$
$968$	1.73205i	0.0556702i
$969$	0	0
$970$	0	0
$971$	31.1769	1.00051	0.500257	−	0.865877i	$-0.333239\pi$
$971$	31.1769	1.00051	0.500257	+	0.865877i	$0.333239\pi$
$972$	0	0
$973$	− 16.0000i	− 0.512936i
$974$	27.7128	0.887976
$975$	0	0
$976$	35.0000	1.12032
$977$	− 48.4974i	− 1.55157i	−0.630997	−	0.775785i	$-0.717354\pi$
$977$	− 48.4974i	− 1.55157i	0.630997	−	0.775785i	$-0.282646\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	− 30.0000i	− 0.957338i
$983$	34.6410i	1.10488i	0.833554	+	0.552438i	$0.186303\pi$
$983$	34.6410i	1.10488i	−0.833554	+	0.552438i	$0.813697\pi$
$984$	0	0
$985$	0	0
$986$	15.5885	0.496438
$987$	0	0
$988$	2.00000i	0.0636285i
$989$	−6.92820	−0.220304
$990$	0	0
$991$	−34.0000	−1.08005	−0.540023	−	0.841650i	$-0.681584\pi$
$991$	−34.0000	−1.08005	−0.540023	+	0.841650i	$0.681584\pi$
$992$	− 41.5692i	− 1.31982i
$993$	0	0
$994$	36.0000	1.14185
$995$	0	0
$996$	0	0
$997$	− 7.00000i	− 0.221692i	−0.993838	−	0.110846i	$-0.964644\pi$
$997$	− 7.00000i	− 0.221692i	0.993838	−	0.110846i	$-0.0353561\pi$
$998$	17.3205i	0.548271i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.k.649.4		4
3.2	odd	2	inner	2025.2.b.k.649.2		4
5.2	odd	4		2025.2.a.j.1.1		2
5.3	odd	4		81.2.a.a.1.2	yes	2
5.4	even	2	inner	2025.2.b.k.649.1		4
15.2	even	4		2025.2.a.j.1.2		2
15.8	even	4		81.2.a.a.1.1	✓	2
15.14	odd	2	inner	2025.2.b.k.649.3		4
20.3	even	4		1296.2.a.o.1.1		2
35.13	even	4		3969.2.a.i.1.2		2
40.3	even	4		5184.2.a.bq.1.2		2
40.13	odd	4		5184.2.a.br.1.2		2
45.13	odd	12		81.2.c.b.55.1		4
45.23	even	12		81.2.c.b.55.2		4
45.38	even	12		81.2.c.b.28.2		4
45.43	odd	12		81.2.c.b.28.1		4
55.43	even	4		9801.2.a.v.1.1		2
60.23	odd	4		1296.2.a.o.1.2		2
105.83	odd	4		3969.2.a.i.1.1		2
120.53	even	4		5184.2.a.br.1.1		2
120.83	odd	4		5184.2.a.bq.1.1		2
135.13	odd	36		729.2.e.o.163.1		12
135.23	even	36		729.2.e.o.406.1		12
135.38	even	36		729.2.e.o.82.1		12
135.43	odd	36		729.2.e.o.82.2		12
135.58	odd	36		729.2.e.o.406.2		12
135.68	even	36		729.2.e.o.163.2		12
135.83	even	36		729.2.e.o.568.2		12
135.88	odd	36		729.2.e.o.325.2		12
135.103	odd	36		729.2.e.o.649.2		12
135.113	even	36		729.2.e.o.649.1		12
135.128	even	36		729.2.e.o.325.1		12
135.133	odd	36		729.2.e.o.568.1		12
165.98	odd	4		9801.2.a.v.1.2		2
180.23	odd	12		1296.2.i.s.865.1		4
180.43	even	12		1296.2.i.s.433.2		4
180.83	odd	12		1296.2.i.s.433.1		4
180.103	even	12		1296.2.i.s.865.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
81.2.a.a.1.1	✓	2	15.8	even	4
81.2.a.a.1.2	yes	2	5.3	odd	4
81.2.c.b.28.1		4	45.43	odd	12
81.2.c.b.28.2		4	45.38	even	12
81.2.c.b.55.1		4	45.13	odd	12
81.2.c.b.55.2		4	45.23	even	12
729.2.e.o.82.1		12	135.38	even	36
729.2.e.o.82.2		12	135.43	odd	36
729.2.e.o.163.1		12	135.13	odd	36
729.2.e.o.163.2		12	135.68	even	36
729.2.e.o.325.1		12	135.128	even	36
729.2.e.o.325.2		12	135.88	odd	36
729.2.e.o.406.1		12	135.23	even	36
729.2.e.o.406.2		12	135.58	odd	36
729.2.e.o.568.1		12	135.133	odd	36
729.2.e.o.568.2		12	135.83	even	36
729.2.e.o.649.1		12	135.113	even	36
729.2.e.o.649.2		12	135.103	odd	36
1296.2.a.o.1.1		2	20.3	even	4
1296.2.a.o.1.2		2	60.23	odd	4
1296.2.i.s.433.1		4	180.83	odd	12
1296.2.i.s.433.2		4	180.43	even	12
1296.2.i.s.865.1		4	180.23	odd	12
1296.2.i.s.865.2		4	180.103	even	12
2025.2.a.j.1.1		2	5.2	odd	4
2025.2.a.j.1.2		2	15.2	even	4
2025.2.b.k.649.1		4	5.4	even	2	inner
2025.2.b.k.649.2		4	3.2	odd	2	inner
2025.2.b.k.649.3		4	15.14	odd	2	inner
2025.2.b.k.649.4		4	1.1	even	1	trivial
3969.2.a.i.1.1		2	105.83	odd	4
3969.2.a.i.1.2		2	35.13	even	4
5184.2.a.bq.1.1		2	120.83	odd	4
5184.2.a.bq.1.2		2	40.3	even	4
5184.2.a.br.1.1		2	120.53	even	4
5184.2.a.br.1.2		2	40.13	odd	4
9801.2.a.v.1.1		2	55.43	even	4
9801.2.a.v.1.2		2	165.98	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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2025.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{2} -1.00000 q^{4} +2.00000i q^{7} +1.73205i q^{8} -3.46410 q^{11} +1.00000i q^{13} -3.46410 q^{14} -5.00000 q^{16} +5.19615i q^{17} -2.00000 q^{19} -6.00000i q^{22} -3.46410i q^{23} -1.73205 q^{26} -2.00000i q^{28} -1.73205 q^{29} +8.00000 q^{31} -5.19615i q^{32} -9.00000 q^{34} -7.00000i q^{37} -3.46410i q^{38} -6.92820 q^{41} -2.00000i q^{43} +3.46410 q^{44} +6.00000 q^{46} +6.92820i q^{47} +3.00000 q^{49} -1.00000i q^{52} -3.46410 q^{56} -3.00000i q^{58} -13.8564 q^{59} -7.00000 q^{61} +13.8564i q^{62} -1.00000 q^{64} -10.0000i q^{67} -5.19615i q^{68} -10.3923 q^{71} +7.00000i q^{73} +12.1244 q^{74} +2.00000 q^{76} -6.92820i q^{77} -2.00000 q^{79} -12.0000i q^{82} +13.8564i q^{83} +3.46410 q^{86} -6.00000i q^{88} +5.19615 q^{89} -2.00000 q^{91} +3.46410i q^{92} -12.0000 q^{94} +2.00000i q^{97} +5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 20 q^{16} - 8 q^{19} + 32 q^{31} - 36 q^{34} + 24 q^{46} + 12 q^{49} - 28 q^{61} - 4 q^{64} + 8 q^{76} - 8 q^{79} - 8 q^{91} - 48 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 - 20 * q^16 - 8 * q^19 + 32 * q^31 - 36 * q^34 + 24 * q^46 + 12 * q^49 - 28 * q^61 - 4 * q^64 + 8 * q^76 - 8 * q^79 - 8 * q^91 - 48 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more