LMFDB - Embedded newform 2025.2.b.j.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:	$\Q(i, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
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			649.4
Root			$2.30278i$ of defining polynomial
Character	$\chi$	$=$	2025.649
Dual form			2025.2.b.j.649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278i q^{2} -3.30278 q^{4} +4.30278i q^{7} -3.00000i q^{8} +O(q^{10})$	$q+2.30278i q^{2} -3.30278 q^{4} +4.30278i q^{7} -3.00000i q^{8} +5.30278 q^{11} +0.302776i q^{13} -9.90833 q^{14} +0.302776 q^{16} +2.30278i q^{17} +7.21110 q^{19} +12.2111i q^{22} +6.90833i q^{23} -0.697224 q^{26} -14.2111i q^{28} +1.39445 q^{29} +8.21110 q^{31} -5.30278i q^{32} -5.30278 q^{34} -1.90833i q^{37} +16.6056i q^{38} -5.30278 q^{41} -5.69722i q^{43} -17.5139 q^{44} -15.9083 q^{46} +6.21110i q^{47} -11.5139 q^{49} -1.00000i q^{52} +5.51388i q^{53} +12.9083 q^{56} +3.21110i q^{58} +3.69722 q^{59} -0.302776 q^{61} +18.9083i q^{62} +12.8167 q^{64} -5.60555i q^{67} -7.60555i q^{68} -4.39445 q^{71} -9.60555i q^{73} +4.39445 q^{74} -23.8167 q^{76} +22.8167i q^{77} +2.60555 q^{79} -12.2111i q^{82} +9.00000i q^{83} +13.1194 q^{86} -15.9083i q^{88} -12.0000 q^{89} -1.30278 q^{91} -22.8167i q^{92} -14.3028 q^{94} +2.00000i q^{97} -26.5139i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{4}+O(q^{10}) $ 4 * q - 6 * q^4	$ 4 q - 6 q^{4} + 14 q^{11} - 18 q^{14} - 6 q^{16} - 10 q^{26} + 20 q^{29} + 4 q^{31} - 14 q^{34} - 14 q^{41} - 34 q^{44} - 42 q^{46} - 10 q^{49} + 30 q^{56} + 22 q^{59} + 6 q^{61} + 8 q^{64} - 32 q^{71} + 32 q^{74} - 52 q^{76} - 4 q^{79} + 2 q^{86} - 48 q^{89} + 2 q^{91} - 50 q^{94}+O(q^{100}) $ 4 * q - 6 * q^4 + 14 * q^11 - 18 * q^14 - 6 * q^16 - 10 * q^26 + 20 * q^29 + 4 * q^31 - 14 * q^34 - 14 * q^41 - 34 * q^44 - 42 * q^46 - 10 * q^49 + 30 * q^56 + 22 * q^59 + 6 * q^61 + 8 * q^64 - 32 * q^71 + 32 * q^74 - 52 * q^76 - 4 * q^79 + 2 * q^86 - 48 * q^89 + 2 * q^91 - 50 * 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 $

$2$	2.30278i	1.62831i	0.580649	+	0.814154i	$0.302799\pi$
$2$	2.30278i	1.62831i	−0.580649	+	0.814154i	$0.697201\pi$
$3$	0	0
$3$	0	0
$4$	−3.30278	−1.65139
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.30278i	1.62630i	0.582057	+	0.813148i	$0.302248\pi$
$7$	4.30278i	1.62630i	−0.582057	+	0.813148i	$0.697752\pi$
$8$	− 3.00000i	− 1.06066i
$9$	0	0
$10$	0	0
$11$	5.30278	1.59885	0.799424	−	0.600768i	$-0.205138\pi$
$11$	5.30278	1.59885	0.799424	+	0.600768i	$0.205138\pi$
$12$	0	0
$13$	0.302776i	0.0839749i	0.999118	+	0.0419874i	$0.0133689\pi$
$13$	0.302776i	0.0839749i	−0.999118	+	0.0419874i	$0.986631\pi$
$14$	−9.90833	−2.64811
$15$	0	0
$16$	0.302776	0.0756939
$17$	2.30278i	0.558505i	0.960218	+	0.279253i	$0.0900867\pi$
$17$	2.30278i	0.558505i	−0.960218	+	0.279253i	$0.909913\pi$
$18$	0	0
$19$	7.21110	1.65434	0.827170	−	0.561951i	$-0.189949\pi$
$19$	7.21110	1.65434	0.827170	+	0.561951i	$0.189949\pi$
$20$	0	0
$21$	0	0
$22$	12.2111i	2.60342i
$23$	6.90833i	1.44049i	0.693722	+	0.720243i	$0.255970\pi$
$23$	6.90833i	1.44049i	−0.693722	+	0.720243i	$0.744030\pi$
$24$	0	0
$25$	0	0
$26$	−0.697224	−0.136737
$27$	0	0
$28$	− 14.2111i	− 2.68565i
$29$	1.39445	0.258943	0.129471	−	0.991583i	$-0.458672\pi$
$29$	1.39445	0.258943	0.129471	+	0.991583i	$0.458672\pi$
$30$	0	0
$31$	8.21110	1.47476	0.737379	−	0.675479i	$-0.236063\pi$
$31$	8.21110	1.47476	0.737379	+	0.675479i	$0.236063\pi$
$32$	− 5.30278i	− 0.937407i
$33$	0	0
$34$	−5.30278	−0.909419
$35$	0	0
$36$	0	0
$37$	− 1.90833i	− 0.313727i	−0.987620	−	0.156864i	$-0.949862\pi$
$37$	− 1.90833i	− 0.313727i	0.987620	−	0.156864i	$-0.0501383\pi$
$38$	16.6056i	2.69378i
$39$	0	0
$40$	0	0
$41$	−5.30278	−0.828154	−0.414077	−	0.910242i	$-0.635896\pi$
$41$	−5.30278	−0.828154	−0.414077	+	0.910242i	$0.635896\pi$
$42$	0	0
$43$	− 5.69722i	− 0.868819i	−0.900716	−	0.434409i	$-0.856957\pi$
$43$	− 5.69722i	− 0.868819i	0.900716	−	0.434409i	$-0.143043\pi$
$44$	−17.5139	−2.64032
$45$	0	0
$46$	−15.9083	−2.34555
$47$	6.21110i	0.905982i	0.891515	+	0.452991i	$0.149643\pi$
$47$	6.21110i	0.905982i	−0.891515	+	0.452991i	$0.850357\pi$
$48$	0	0
$49$	−11.5139	−1.64484
$50$	0	0
$51$	0	0
$52$	− 1.00000i	− 0.138675i
$53$	5.51388i	0.757389i	0.925522	+	0.378695i	$0.123627\pi$
$53$	5.51388i	0.757389i	−0.925522	+	0.378695i	$0.876373\pi$
$54$	0	0
$55$	0	0
$56$	12.9083	1.72495
$57$	0	0
$58$	3.21110i	0.421638i
$59$	3.69722	0.481338	0.240669	−	0.970607i	$-0.422633\pi$
$59$	3.69722	0.481338	0.240669	+	0.970607i	$0.422633\pi$
$60$	0	0
$61$	−0.302776	−0.0387664	−0.0193832	−	0.999812i	$-0.506170\pi$
$61$	−0.302776	−0.0387664	−0.0193832	+	0.999812i	$0.506170\pi$
$62$	18.9083i	2.40136i
$63$	0	0
$64$	12.8167	1.60208
$65$	0	0
$66$	0	0
$67$	− 5.60555i	− 0.684827i	−0.939549	−	0.342414i	$-0.888756\pi$
$67$	− 5.60555i	− 0.684827i	0.939549	−	0.342414i	$-0.111244\pi$
$68$	− 7.60555i	− 0.922309i
$69$	0	0
$70$	0	0
$71$	−4.39445	−0.521525	−0.260763	−	0.965403i	$-0.583974\pi$
$71$	−4.39445	−0.521525	−0.260763	+	0.965403i	$0.583974\pi$
$72$	0	0
$73$	− 9.60555i	− 1.12424i	−0.827054	−	0.562122i	$-0.809985\pi$
$73$	− 9.60555i	− 1.12424i	0.827054	−	0.562122i	$-0.190015\pi$
$74$	4.39445	0.510844
$75$	0	0
$76$	−23.8167	−2.73196
$77$	22.8167i	2.60020i
$78$	0	0
$79$	2.60555	0.293147	0.146574	−	0.989200i	$-0.453175\pi$
$79$	2.60555	0.293147	0.146574	+	0.989200i	$0.453175\pi$
$80$	0	0
$81$	0	0
$82$	− 12.2111i	− 1.34849i
$83$	9.00000i	0.987878i	0.869496	+	0.493939i	$0.164443\pi$
$83$	9.00000i	0.987878i	−0.869496	+	0.493939i	$0.835557\pi$
$84$	0	0
$85$	0	0
$86$	13.1194	1.41470
$87$	0	0
$88$	− 15.9083i	− 1.69583i
$89$	−12.0000	−1.27200	−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000	−1.27200	−0.635999	+	0.771690i	$0.719412\pi$
$90$	0	0
$91$	−1.30278	−0.136568
$92$	− 22.8167i	− 2.37880i
$93$	0	0
$94$	−14.3028	−1.47522
$95$	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	$0.967625\pi$
$98$	− 26.5139i	− 2.67831i
$99$	0	0
$100$	0	0
$101$	−14.7250	−1.46519	−0.732595	−	0.680665i	$-0.761691\pi$
$101$	−14.7250	−1.46519	−0.732595	+	0.680665i	$0.761691\pi$
$102$	0	0
$103$	− 17.9083i	− 1.76456i	−0.470725	−	0.882280i	$-0.656008\pi$
$103$	− 17.9083i	− 1.76456i	0.470725	−	0.882280i	$-0.343992\pi$
$104$	0.908327	0.0890688
$105$	0	0
$106$	−12.6972	−1.23326
$107$	− 9.00000i	− 0.870063i	−0.900415	−	0.435031i	$-0.856737\pi$
$107$	− 9.00000i	− 0.870063i	0.900415	−	0.435031i	$-0.143263\pi$
$108$	0	0
$109$	−11.9083	−1.14061	−0.570305	−	0.821433i	$-0.693175\pi$
$109$	−11.9083	−1.14061	−0.570305	+	0.821433i	$0.693175\pi$
$110$	0	0
$111$	0	0
$112$	1.30278i	0.123101i
$113$	− 11.3028i	− 1.06328i	−0.846972	−	0.531638i	$-0.821577\pi$
$113$	− 11.3028i	− 1.06328i	0.846972	−	0.531638i	$-0.178423\pi$
$114$	0	0
$115$	0	0
$116$	−4.60555	−0.427615
$117$	0	0
$118$	8.51388i	0.783766i
$119$	−9.90833	−0.908295
$120$	0	0
$121$	17.1194	1.55631
$122$	− 0.697224i	− 0.0631237i
$123$	0	0
$124$	−27.1194	−2.43540
$125$	0	0
$126$	0	0
$127$	3.60555i	0.319941i	0.987122	+	0.159970i	$0.0511399\pi$
$127$	3.60555i	0.319941i	−0.987122	+	0.159970i	$0.948860\pi$
$128$	18.9083i	1.67128i
$129$	0	0
$130$	0	0
$131$	−1.81665	−0.158722	−0.0793609	−	0.996846i	$-0.525288\pi$
$131$	−1.81665	−0.158722	−0.0793609	+	0.996846i	$0.525288\pi$
$132$	0	0
$133$	31.0278i	2.69045i
$134$	12.9083	1.11511
$135$	0	0
$136$	6.90833	0.592384
$137$	− 12.9083i	− 1.10283i	−0.834230	−	0.551416i	$-0.814088\pi$
$137$	− 12.9083i	− 1.10283i	0.834230	−	0.551416i	$-0.185912\pi$
$138$	0	0
$139$	7.90833	0.670776	0.335388	−	0.942080i	$-0.391133\pi$
$139$	7.90833	0.670776	0.335388	+	0.942080i	$0.391133\pi$
$140$	0	0
$141$	0	0
$142$	− 10.1194i	− 0.849204i
$143$	1.60555i	0.134263i
$144$	0	0
$145$	0	0
$146$	22.1194	1.83062
$147$	0	0
$148$	6.30278i	0.518085i
$149$	6.21110	0.508833	0.254417	−	0.967095i	$-0.418117\pi$
$149$	6.21110	0.508833	0.254417	+	0.967095i	$0.418117\pi$
$150$	0	0
$151$	−23.1194	−1.88143	−0.940716	−	0.339195i	$-0.889846\pi$
$151$	−23.1194	−1.88143	−0.940716	+	0.339195i	$0.889846\pi$
$152$	− 21.6333i	− 1.75469i
$153$	0	0
$154$	−52.5416	−4.23393
$155$	0	0
$156$	0	0
$157$	3.18335i	0.254059i	0.991899	+	0.127029i	$0.0405442\pi$
$157$	3.18335i	0.254059i	−0.991899	+	0.127029i	$0.959456\pi$
$158$	6.00000i	0.477334i
$159$	0	0
$160$	0	0
$161$	−29.7250	−2.34266
$162$	0	0
$163$	2.81665i	0.220617i	0.993897	+	0.110309i	$0.0351839\pi$
$163$	2.81665i	0.220617i	−0.993897	+	0.110309i	$0.964816\pi$
$164$	17.5139	1.36760
$165$	0	0
$166$	−20.7250	−1.60857
$167$	− 0.908327i	− 0.0702884i	−0.999382	−	0.0351442i	$-0.988811\pi$
$167$	− 0.908327i	− 0.0702884i	0.999382	−	0.0351442i	$-0.0111891\pi$
$168$	0	0
$169$	12.9083	0.992948
$170$	0	0
$171$	0	0
$172$	18.8167i	1.43476i
$173$	− 7.81665i	− 0.594289i	−0.954832	−	0.297145i	$-0.903966\pi$
$173$	− 7.81665i	− 0.594289i	0.954832	−	0.297145i	$-0.0960343\pi$
$174$	0	0
$175$	0	0
$176$	1.60555	0.121023
$177$	0	0
$178$	− 27.6333i	− 2.07120i
$179$	15.4222	1.15271	0.576355	−	0.817200i	$-0.304475\pi$
$179$	15.4222	1.15271	0.576355	+	0.817200i	$0.304475\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	− 3.00000i	− 0.222375i
$183$	0	0
$184$	20.7250	1.52787
$185$	0	0
$186$	0	0
$187$	12.2111i	0.892964i
$188$	− 20.5139i	− 1.49613i
$189$	0	0
$190$	0	0
$191$	−9.21110	−0.666492	−0.333246	−	0.942840i	$-0.608144\pi$
$191$	−9.21110	−0.666492	−0.333246	+	0.942840i	$0.608144\pi$
$192$	0	0
$193$	− 6.81665i	− 0.490673i	−0.969438	−	0.245337i	$-0.921101\pi$
$193$	− 6.81665i	− 0.490673i	0.969438	−	0.245337i	$-0.0788985\pi$
$194$	−4.60555	−0.330659
$195$	0	0
$196$	38.0278	2.71627
$197$	− 2.09167i	− 0.149026i	−0.997220	−	0.0745128i	$-0.976260\pi$
$197$	− 2.09167i	− 0.149026i	0.997220	−	0.0745128i	$-0.0237402\pi$
$198$	0	0
$199$	7.48612	0.530677	0.265339	−	0.964155i	$-0.414516\pi$
$199$	7.48612	0.530677	0.265339	+	0.964155i	$0.414516\pi$
$200$	0	0
$201$	0	0
$202$	− 33.9083i	− 2.38578i
$203$	6.00000i	0.421117i
$204$	0	0
$205$	0	0
$206$	41.2389	2.87325
$207$	0	0
$208$	0.0916731i	0.00635638i
$209$	38.2389	2.64504
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	− 18.2111i	− 1.25074i
$213$	0	0
$214$	20.7250	1.41673
$215$	0	0
$216$	0	0
$217$	35.3305i	2.39839i
$218$	− 27.4222i	− 1.85727i
$219$	0	0
$220$	0	0
$221$	−0.697224	−0.0469004
$222$	0	0
$223$	− 17.0000i	− 1.13840i	−0.822198	−	0.569202i	$-0.807252\pi$
$223$	− 17.0000i	− 1.13840i	0.822198	−	0.569202i	$-0.192748\pi$
$224$	22.8167	1.52450
$225$	0	0
$226$	26.0278	1.73134
$227$	25.6056i	1.69950i	0.527186	+	0.849750i	$0.323247\pi$
$227$	25.6056i	1.69950i	−0.527186	+	0.849750i	$0.676753\pi$
$228$	0	0
$229$	19.4222	1.28346	0.641728	−	0.766933i	$-0.278218\pi$
$229$	19.4222	1.28346	0.641728	+	0.766933i	$0.278218\pi$
$230$	0	0
$231$	0	0
$232$	− 4.18335i	− 0.274650i
$233$	7.18335i	0.470597i	0.971923	+	0.235298i	$0.0756067\pi$
$233$	7.18335i	0.470597i	−0.971923	+	0.235298i	$0.924393\pi$
$234$	0	0
$235$	0	0
$236$	−12.2111	−0.794875
$237$	0	0
$238$	− 22.8167i	− 1.47898i
$239$	24.4222	1.57974	0.789871	−	0.613274i	$-0.210148\pi$
$239$	24.4222	1.57974	0.789871	+	0.613274i	$0.210148\pi$
$240$	0	0
$241$	4.78890	0.308480	0.154240	−	0.988033i	$-0.450707\pi$
$241$	4.78890	0.308480	0.154240	+	0.988033i	$0.450707\pi$
$242$	39.4222i	2.53416i
$243$	0	0
$244$	1.00000	0.0640184
$245$	0	0
$246$	0	0
$247$	2.18335i	0.138923i
$248$	− 24.6333i	− 1.56422i
$249$	0	0
$250$	0	0
$251$	−26.7250	−1.68687	−0.843433	−	0.537235i	$-0.819469\pi$
$251$	−26.7250	−1.68687	−0.843433	+	0.537235i	$0.819469\pi$
$252$	0	0
$253$	36.6333i	2.30312i
$254$	−8.30278	−0.520962
$255$	0	0
$256$	−17.9083	−1.11927
$257$	− 21.6333i	− 1.34945i	−0.738070	−	0.674724i	$-0.764263\pi$
$257$	− 21.6333i	− 1.34945i	0.738070	−	0.674724i	$-0.235737\pi$
$258$	0	0
$259$	8.21110	0.510213
$260$	0	0
$261$	0	0
$262$	− 4.18335i	− 0.258448i
$263$	11.0278i	0.680001i	0.940425	+	0.340000i	$0.110427\pi$
$263$	11.0278i	0.680001i	−0.940425	+	0.340000i	$0.889573\pi$
$264$	0	0
$265$	0	0
$266$	−71.4500	−4.38088
$267$	0	0
$268$	18.5139i	1.13092i
$269$	−17.3028	−1.05497	−0.527484	−	0.849565i	$-0.676865\pi$
$269$	−17.3028	−1.05497	−0.527484	+	0.849565i	$0.676865\pi$
$270$	0	0
$271$	9.60555	0.583496	0.291748	−	0.956495i	$-0.405763\pi$
$271$	9.60555	0.583496	0.291748	+	0.956495i	$0.405763\pi$
$272$	0.697224i	0.0422754i
$273$	0	0
$274$	29.7250	1.79575
$275$	0	0
$276$	0	0
$277$	29.2111i	1.75513i	0.479462	+	0.877563i	$0.340832\pi$
$277$	29.2111i	1.75513i	−0.479462	+	0.877563i	$0.659168\pi$
$278$	18.2111i	1.09223i
$279$	0	0
$280$	0	0
$281$	4.81665	0.287337	0.143669	−	0.989626i	$-0.454110\pi$
$281$	4.81665	0.287337	0.143669	+	0.989626i	$0.454110\pi$
$282$	0	0
$283$	− 15.1194i	− 0.898757i	−0.893342	−	0.449378i	$-0.851646\pi$
$283$	− 15.1194i	− 0.898757i	0.893342	−	0.449378i	$-0.148354\pi$
$284$	14.5139	0.861240
$285$	0	0
$286$	−3.69722	−0.218621
$287$	− 22.8167i	− 1.34682i
$288$	0	0
$289$	11.6972	0.688072
$290$	0	0
$291$	0	0
$292$	31.7250i	1.85656i
$293$	− 1.18335i	− 0.0691318i	−0.999402	−	0.0345659i	$-0.988995\pi$
$293$	− 1.18335i	− 0.0691318i	0.999402	−	0.0345659i	$-0.0110049\pi$
$294$	0	0
$295$	0	0
$296$	−5.72498	−0.332758
$297$	0	0
$298$	14.3028i	0.828538i
$299$	−2.09167	−0.120965
$300$	0	0
$301$	24.5139	1.41296
$302$	− 53.2389i	− 3.06355i
$303$	0	0
$304$	2.18335	0.125223
$305$	0	0
$306$	0	0
$307$	4.78890i	0.273317i	0.990618	+	0.136658i	$0.0436363\pi$
$307$	4.78890i	0.273317i	−0.990618	+	0.136658i	$0.956364\pi$
$308$	− 75.3583i	− 4.29394i
$309$	0	0
$310$	0	0
$311$	21.0000	1.19080	0.595400	−	0.803429i	$-0.296993\pi$
$311$	21.0000	1.19080	0.595400	+	0.803429i	$0.296993\pi$
$312$	0	0
$313$	− 10.7250i	− 0.606212i	−0.952957	−	0.303106i	$-0.901976\pi$
$313$	− 10.7250i	− 0.606212i	0.952957	−	0.303106i	$-0.0980236\pi$
$314$	−7.33053	−0.413686
$315$	0	0
$316$	−8.60555	−0.484100
$317$	21.4222i	1.20319i	0.798801	+	0.601595i	$0.205468\pi$
$317$	21.4222i	1.20319i	−0.798801	+	0.601595i	$0.794532\pi$
$318$	0	0
$319$	7.39445	0.414010
$320$	0	0
$321$	0	0
$322$	− 68.4500i	− 3.81457i
$323$	16.6056i	0.923958i
$324$	0	0
$325$	0	0
$326$	−6.48612	−0.359233
$327$	0	0
$328$	15.9083i	0.878390i
$329$	−26.7250	−1.47340
$330$	0	0
$331$	−16.4222	−0.902646	−0.451323	−	0.892361i	$-0.649048\pi$
$331$	−16.4222	−0.902646	−0.451323	+	0.892361i	$0.649048\pi$
$332$	− 29.7250i	− 1.63137i
$333$	0	0
$334$	2.09167	0.114451
$335$	0	0
$336$	0	0
$337$	− 11.3944i	− 0.620695i	−0.950623	−	0.310348i	$-0.899555\pi$
$337$	− 11.3944i	− 0.620695i	0.950623	−	0.310348i	$-0.100445\pi$
$338$	29.7250i	1.61683i
$339$	0	0
$340$	0	0
$341$	43.5416	2.35791
$342$	0	0
$343$	− 19.4222i	− 1.04870i
$344$	−17.0917	−0.921521
$345$	0	0
$346$	18.0000	0.967686
$347$	26.9361i	1.44600i	0.690846	+	0.723002i	$0.257238\pi$
$347$	26.9361i	1.44600i	−0.690846	+	0.723002i	$0.742762\pi$
$348$	0	0
$349$	5.39445	0.288758	0.144379	−	0.989522i	$-0.453882\pi$
$349$	5.39445	0.288758	0.144379	+	0.989522i	$0.453882\pi$
$350$	0	0
$351$	0	0
$352$	− 28.1194i	− 1.49877i
$353$	17.7250i	0.943406i	0.881758	+	0.471703i	$0.156360\pi$
$353$	17.7250i	0.943406i	−0.881758	+	0.471703i	$0.843640\pi$
$354$	0	0
$355$	0	0
$356$	39.6333	2.10056
$357$	0	0
$358$	35.5139i	1.87697i
$359$	9.90833	0.522941	0.261471	−	0.965211i	$-0.415793\pi$
$359$	9.90833	0.522941	0.261471	+	0.965211i	$0.415793\pi$
$360$	0	0
$361$	33.0000	1.73684
$362$	− 16.1194i	− 0.847218i
$363$	0	0
$364$	4.30278	0.225527
$365$	0	0
$366$	0	0
$367$	13.3028i	0.694399i	0.937791	+	0.347200i	$0.112867\pi$
$367$	13.3028i	0.694399i	−0.937791	+	0.347200i	$0.887133\pi$
$368$	2.09167i	0.109036i
$369$	0	0
$370$	0	0
$371$	−23.7250	−1.23174
$372$	0	0
$373$	8.39445i	0.434648i	0.976100	+	0.217324i	$0.0697328\pi$
$373$	8.39445i	0.434648i	−0.976100	+	0.217324i	$0.930267\pi$
$374$	−28.1194	−1.45402
$375$	0	0
$376$	18.6333	0.960939
$377$	0.422205i	0.0217447i
$378$	0	0
$379$	−5.69722	−0.292647	−0.146323	−	0.989237i	$-0.546744\pi$
$379$	−5.69722	−0.292647	−0.146323	+	0.989237i	$0.546744\pi$
$380$	0	0
$381$	0	0
$382$	− 21.2111i	− 1.08525i
$383$	− 5.72498i	− 0.292533i	−0.989245	−	0.146266i	$-0.953274\pi$
$383$	− 5.72498i	− 0.292533i	0.989245	−	0.146266i	$-0.0467257\pi$
$384$	0	0
$385$	0	0
$386$	15.6972	0.798968
$387$	0	0
$388$	− 6.60555i	− 0.335346i
$389$	1.18335	0.0599980	0.0299990	−	0.999550i	$-0.490450\pi$
$389$	1.18335	0.0599980	0.0299990	+	0.999550i	$0.490450\pi$
$390$	0	0
$391$	−15.9083	−0.804519
$392$	34.5416i	1.74462i
$393$	0	0
$394$	4.81665	0.242660
$395$	0	0
$396$	0	0
$397$	− 14.3944i	− 0.722437i	−0.932481	−	0.361218i	$-0.882361\pi$
$397$	− 14.3944i	− 0.722437i	0.932481	−	0.361218i	$-0.117639\pi$
$398$	17.2389i	0.864106i
$399$	0	0
$400$	0	0
$401$	4.18335	0.208906	0.104453	−	0.994530i	$-0.466691\pi$
$401$	4.18335	0.208906	0.104453	+	0.994530i	$0.466691\pi$
$402$	0	0
$403$	2.48612i	0.123843i
$404$	48.6333	2.41960
$405$	0	0
$406$	−13.8167	−0.685709
$407$	− 10.1194i	− 0.501601i
$408$	0	0
$409$	−30.5416	−1.51019	−0.755093	−	0.655617i	$-0.772408\pi$
$409$	−30.5416	−1.51019	−0.755093	+	0.655617i	$0.772408\pi$
$410$	0	0
$411$	0	0
$412$	59.1472i	2.91397i
$413$	15.9083i	0.782798i
$414$	0	0
$415$	0	0
$416$	1.60555	0.0787186
$417$	0	0
$418$	88.0555i	4.30694i
$419$	35.3028	1.72465	0.862327	−	0.506352i	$-0.169006\pi$
$419$	35.3028	1.72465	0.862327	+	0.506352i	$0.169006\pi$
$420$	0	0
$421$	12.6056	0.614357	0.307178	−	0.951652i	$-0.400615\pi$
$421$	12.6056	0.614357	0.307178	+	0.951652i	$0.400615\pi$
$422$	11.5139i	0.560487i
$423$	0	0
$424$	16.5416	0.803333
$425$	0	0
$426$	0	0
$427$	− 1.30278i	− 0.0630457i
$428$	29.7250i	1.43681i
$429$	0	0
$430$	0	0
$431$	−10.8167	−0.521020	−0.260510	−	0.965471i	$-0.583891\pi$
$431$	−10.8167	−0.521020	−0.260510	+	0.965471i	$0.583891\pi$
$432$	0	0
$433$	− 16.2389i	− 0.780390i	−0.920732	−	0.390195i	$-0.872408\pi$
$433$	− 16.2389i	− 0.780390i	0.920732	−	0.390195i	$-0.127592\pi$
$434$	−81.3583	−3.90532
$435$	0	0
$436$	39.3305	1.88359
$437$	49.8167i	2.38305i
$438$	0	0
$439$	13.4861	0.643657	0.321829	−	0.946798i	$-0.395702\pi$
$439$	13.4861	0.643657	0.321829	+	0.946798i	$0.395702\pi$
$440$	0	0
$441$	0	0
$442$	− 1.60555i	− 0.0763683i
$443$	− 29.0917i	− 1.38219i	−0.722765	−	0.691094i	$-0.757129\pi$
$443$	− 29.0917i	− 1.38219i	0.722765	−	0.691094i	$-0.242871\pi$
$444$	0	0
$445$	0	0
$446$	39.1472	1.85367
$447$	0	0
$448$	55.1472i	2.60546i
$449$	1.88057	0.0887496	0.0443748	−	0.999015i	$-0.485870\pi$
$449$	1.88057	0.0887496	0.0443748	+	0.999015i	$0.485870\pi$
$450$	0	0
$451$	−28.1194	−1.32409
$452$	37.3305i	1.75588i
$453$	0	0
$454$	−58.9638	−2.76731
$455$	0	0
$456$	0	0
$457$	− 3.57779i	− 0.167362i	−0.996493	−	0.0836811i	$-0.973332\pi$
$457$	− 3.57779i	− 0.167362i	0.996493	−	0.0836811i	$-0.0266677\pi$
$458$	44.7250i	2.08986i
$459$	0	0
$460$	0	0
$461$	23.7250	1.10498	0.552491	−	0.833519i	$-0.313677\pi$
$461$	23.7250	1.10498	0.552491	+	0.833519i	$0.313677\pi$
$462$	0	0
$463$	17.3305i	0.805418i	0.915328	+	0.402709i	$0.131931\pi$
$463$	17.3305i	0.805418i	−0.915328	+	0.402709i	$0.868069\pi$
$464$	0.422205	0.0196004
$465$	0	0
$466$	−16.5416	−0.766276
$467$	− 5.57779i	− 0.258110i	−0.991637	−	0.129055i	$-0.958806\pi$
$467$	− 5.57779i	− 0.258110i	0.991637	−	0.129055i	$-0.0411943\pi$
$468$	0	0
$469$	24.1194	1.11373
$470$	0	0
$471$	0	0
$472$	− 11.0917i	− 0.510536i
$473$	− 30.2111i	− 1.38911i
$474$	0	0
$475$	0	0
$476$	32.7250	1.49995
$477$	0	0
$478$	56.2389i	2.57231i
$479$	−33.9083	−1.54931	−0.774656	−	0.632383i	$-0.782077\pi$
$479$	−33.9083	−1.54931	−0.774656	+	0.632383i	$0.782077\pi$
$480$	0	0
$481$	0.577795	0.0263452
$482$	11.0278i	0.502301i
$483$	0	0
$484$	−56.5416	−2.57007
$485$	0	0
$486$	0	0
$487$	− 7.63331i	− 0.345898i	−0.984931	−	0.172949i	$-0.944670\pi$
$487$	− 7.63331i	− 0.345898i	0.984931	−	0.172949i	$-0.0553296\pi$
$488$	0.908327i	0.0411180i
$489$	0	0
$490$	0	0
$491$	23.2389	1.04876	0.524378	−	0.851486i	$-0.324298\pi$
$491$	23.2389	1.04876	0.524378	+	0.851486i	$0.324298\pi$
$492$	0	0
$493$	3.21110i	0.144621i
$494$	−5.02776	−0.226209
$495$	0	0
$496$	2.48612	0.111630
$497$	− 18.9083i	− 0.848154i
$498$	0	0
$499$	−33.3305	−1.49208	−0.746040	−	0.665901i	$-0.768047\pi$
$499$	−33.3305	−1.49208	−0.746040	+	0.665901i	$0.768047\pi$
$500$	0	0
$501$	0	0
$502$	− 61.5416i	− 2.74674i
$503$	− 25.3305i	− 1.12943i	−0.825285	−	0.564716i	$-0.808986\pi$
$503$	− 25.3305i	− 1.12943i	0.825285	−	0.564716i	$-0.191014\pi$
$504$	0	0
$505$	0	0
$506$	−84.3583	−3.75018
$507$	0	0
$508$	− 11.9083i	− 0.528347i
$509$	5.78890	0.256588	0.128294	−	0.991736i	$-0.459050\pi$
$509$	5.78890	0.256588	0.128294	+	0.991736i	$0.459050\pi$
$510$	0	0
$511$	41.3305	1.82836
$512$	− 3.42221i	− 0.151242i
$513$	0	0
$514$	49.8167	2.19732
$515$	0	0
$516$	0	0
$517$	32.9361i	1.44853i
$518$	18.9083i	0.830784i
$519$	0	0
$520$	0	0
$521$	−41.0278	−1.79746	−0.898729	−	0.438504i	$-0.855509\pi$
$521$	−41.0278	−1.79746	−0.898729	+	0.438504i	$0.855509\pi$
$522$	0	0
$523$	− 37.2389i	− 1.62834i	−0.580625	−	0.814171i	$-0.697192\pi$
$523$	− 37.2389i	− 1.62834i	0.580625	−	0.814171i	$-0.302808\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	−25.3944	−1.10725
$527$	18.9083i	0.823660i
$528$	0	0
$529$	−24.7250	−1.07500
$530$	0	0
$531$	0	0
$532$	− 102.478i	− 4.44297i
$533$	− 1.60555i	− 0.0695441i
$534$	0	0
$535$	0	0
$536$	−16.8167	−0.726369
$537$	0	0
$538$	− 39.8444i	− 1.71781i
$539$	−61.0555	−2.62985
$540$	0	0
$541$	−20.3305	−0.874078	−0.437039	−	0.899443i	$-0.643973\pi$
$541$	−20.3305	−0.874078	−0.437039	+	0.899443i	$0.643973\pi$
$542$	22.1194i	0.950111i
$543$	0	0
$544$	12.2111	0.523547
$545$	0	0
$546$	0	0
$547$	− 39.6611i	− 1.69578i	−0.530168	−	0.847892i	$-0.677871\pi$
$547$	− 39.6611i	− 1.69578i	0.530168	−	0.847892i	$-0.322129\pi$
$548$	42.6333i	1.82120i
$549$	0	0
$550$	0	0
$551$	10.0555	0.428379
$552$	0	0
$553$	11.2111i	0.476745i
$554$	−67.2666	−2.85788
$555$	0	0
$556$	−26.1194	−1.10771
$557$	− 33.6333i	− 1.42509i	−0.701627	−	0.712544i	$-0.747543\pi$
$557$	− 33.6333i	− 1.42509i	0.701627	−	0.712544i	$-0.252457\pi$
$558$	0	0
$559$	1.72498	0.0729589
$560$	0	0
$561$	0	0
$562$	11.0917i	0.467874i
$563$	36.6333i	1.54391i	0.635677	+	0.771955i	$0.280721\pi$
$563$	36.6333i	1.54391i	−0.635677	+	0.771955i	$0.719279\pi$
$564$	0	0
$565$	0	0
$566$	34.8167	1.46345
$567$	0	0
$568$	13.1833i	0.553161i
$569$	−19.8167	−0.830757	−0.415379	−	0.909649i	$-0.636351\pi$
$569$	−19.8167	−0.830757	−0.415379	+	0.909649i	$0.636351\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	− 5.30278i	− 0.221720i
$573$	0	0
$574$	52.5416	2.19305
$575$	0	0
$576$	0	0
$577$	− 10.6333i	− 0.442670i	−0.975198	−	0.221335i	$-0.928959\pi$
$577$	− 10.6333i	− 0.442670i	0.975198	−	0.221335i	$-0.0710415\pi$
$578$	26.9361i	1.12039i
$579$	0	0
$580$	0	0
$581$	−38.7250	−1.60658
$582$	0	0
$583$	29.2389i	1.21095i
$584$	−28.8167	−1.19244
$585$	0	0
$586$	2.72498	0.112568
$587$	− 4.81665i	− 0.198805i	−0.995047	−	0.0994023i	$-0.968307\pi$
$587$	− 4.81665i	− 0.198805i	0.995047	−	0.0994023i	$-0.0316931\pi$
$588$	0	0
$589$	59.2111	2.43975
$590$	0	0
$591$	0	0
$592$	− 0.577795i	− 0.0237472i
$593$	37.1194i	1.52431i	0.647393	+	0.762156i	$0.275859\pi$
$593$	37.1194i	1.52431i	−0.647393	+	0.762156i	$0.724141\pi$
$594$	0	0
$595$	0	0
$596$	−20.5139	−0.840281
$597$	0	0
$598$	− 4.81665i	− 0.196968i
$599$	−19.3305	−0.789824	−0.394912	−	0.918719i	$-0.629225\pi$
$599$	−19.3305	−0.789824	−0.394912	+	0.918719i	$0.629225\pi$
$600$	0	0
$601$	17.9083	0.730496	0.365248	−	0.930910i	$-0.380984\pi$
$601$	17.9083	0.730496	0.365248	+	0.930910i	$0.380984\pi$
$602$	56.4500i	2.30073i
$603$	0	0
$604$	76.3583	3.10697
$605$	0	0
$606$	0	0
$607$	28.2389i	1.14618i	0.819492	+	0.573090i	$0.194255\pi$
$607$	28.2389i	1.14618i	−0.819492	+	0.573090i	$0.805745\pi$
$608$	− 38.2389i	− 1.55079i
$609$	0	0
$610$	0	0
$611$	−1.88057	−0.0760797
$612$	0	0
$613$	17.3944i	0.702555i	0.936271	+	0.351278i	$0.114253\pi$
$613$	17.3944i	0.702555i	−0.936271	+	0.351278i	$0.885747\pi$
$614$	−11.0278	−0.445044
$615$	0	0
$616$	68.4500	2.75793
$617$	4.81665i	0.193911i	0.995289	+	0.0969556i	$0.0309105\pi$
$617$	4.81665i	0.193911i	−0.995289	+	0.0969556i	$0.969090\pi$
$618$	0	0
$619$	−14.0000	−0.562708	−0.281354	−	0.959604i	$-0.590783\pi$
$619$	−14.0000	−0.562708	−0.281354	+	0.959604i	$0.590783\pi$
$620$	0	0
$621$	0	0
$622$	48.3583i	1.93899i
$623$	− 51.6333i	− 2.06864i
$624$	0	0
$625$	0	0
$626$	24.6972	0.987100
$627$	0	0
$628$	− 10.5139i	− 0.419549i
$629$	4.39445	0.175218
$630$	0	0
$631$	−3.30278	−0.131481	−0.0657407	−	0.997837i	$-0.520941\pi$
$631$	−3.30278	−0.131481	−0.0657407	+	0.997837i	$0.520941\pi$
$632$	− 7.81665i	− 0.310930i
$633$	0	0
$634$	−49.3305	−1.95917
$635$	0	0
$636$	0	0
$637$	− 3.48612i	− 0.138125i
$638$	17.0278i	0.674135i
$639$	0	0
$640$	0	0
$641$	32.0917	1.26754	0.633772	−	0.773520i	$-0.281506\pi$
$641$	32.0917	1.26754	0.633772	+	0.773520i	$0.281506\pi$
$642$	0	0
$643$	− 5.48612i	− 0.216352i	−0.994132	−	0.108176i	$-0.965499\pi$
$643$	− 5.48612i	− 0.216352i	0.994132	−	0.108176i	$-0.0345009\pi$
$644$	98.1749	3.86863
$645$	0	0
$646$	−38.2389	−1.50449
$647$	33.8444i	1.33056i	0.746593	+	0.665281i	$0.231688\pi$
$647$	33.8444i	1.33056i	−0.746593	+	0.665281i	$0.768312\pi$
$648$	0	0
$649$	19.6056	0.769585
$650$	0	0
$651$	0	0
$652$	− 9.30278i	− 0.364325i
$653$	1.11943i	0.0438067i	0.999760	+	0.0219033i	$0.00697260\pi$
$653$	1.11943i	0.0438067i	−0.999760	+	0.0219033i	$0.993027\pi$
$654$	0	0
$655$	0	0
$656$	−1.60555	−0.0626862
$657$	0	0
$658$	− 61.5416i	− 2.39914i
$659$	−22.1194	−0.861651	−0.430825	−	0.902435i	$-0.641777\pi$
$659$	−22.1194	−0.861651	−0.430825	+	0.902435i	$0.641777\pi$
$660$	0	0
$661$	−10.2111	−0.397166	−0.198583	−	0.980084i	$-0.563634\pi$
$661$	−10.2111	−0.397166	−0.198583	+	0.980084i	$0.563634\pi$
$662$	− 37.8167i	− 1.46979i
$663$	0	0
$664$	27.0000	1.04780
$665$	0	0
$666$	0	0
$667$	9.63331i	0.373003i
$668$	3.00000i	0.116073i
$669$	0	0
$670$	0	0
$671$	−1.60555	−0.0619816
$672$	0	0
$673$	− 16.0278i	− 0.617825i	−0.951091	−	0.308912i	$-0.900035\pi$
$673$	− 16.0278i	− 0.617825i	0.951091	−	0.308912i	$-0.0999650\pi$
$674$	26.2389	1.01068
$675$	0	0
$676$	−42.6333	−1.63974
$677$	3.00000i	0.115299i	0.998337	+	0.0576497i	$0.0183606\pi$
$677$	3.00000i	0.115299i	−0.998337	+	0.0576497i	$0.981639\pi$
$678$	0	0
$679$	−8.60555	−0.330251
$680$	0	0
$681$	0	0
$682$	100.267i	3.83941i
$683$	− 32.7250i	− 1.25219i	−0.779748	−	0.626093i	$-0.784653\pi$
$683$	− 32.7250i	− 1.25219i	0.779748	−	0.626093i	$-0.215347\pi$
$684$	0	0
$685$	0	0
$686$	44.7250	1.70761
$687$	0	0
$688$	− 1.72498i	− 0.0657643i
$689$	−1.66947	−0.0636017
$690$	0	0
$691$	33.0555	1.25749	0.628745	−	0.777611i	$-0.283569\pi$
$691$	33.0555	1.25749	0.628745	+	0.777611i	$0.283569\pi$
$692$	25.8167i	0.981402i
$693$	0	0
$694$	−62.0278	−2.35454
$695$	0	0
$696$	0	0
$697$	− 12.2111i	− 0.462528i
$698$	12.4222i	0.470187i
$699$	0	0
$700$	0	0
$701$	50.2389	1.89750	0.948748	−	0.316034i	$-0.102351\pi$
$701$	50.2389	1.89750	0.948748	+	0.316034i	$0.102351\pi$
$702$	0	0
$703$	− 13.7611i	− 0.519011i
$704$	67.9638	2.56148
$705$	0	0
$706$	−40.8167	−1.53616
$707$	− 63.3583i	− 2.38283i
$708$	0	0
$709$	28.2111	1.05949	0.529745	−	0.848157i	$-0.322288\pi$
$709$	28.2111	1.05949	0.529745	+	0.848157i	$0.322288\pi$
$710$	0	0
$711$	0	0
$712$	36.0000i	1.34916i
$713$	56.7250i	2.12437i
$714$	0	0
$715$	0	0
$716$	−50.9361	−1.90357
$717$	0	0
$718$	22.8167i	0.851510i
$719$	−24.9083	−0.928924	−0.464462	−	0.885593i	$-0.653752\pi$
$719$	−24.9083	−0.928924	−0.464462	+	0.885593i	$0.653752\pi$
$720$	0	0
$721$	77.0555	2.86970
$722$	75.9916i	2.82811i
$723$	0	0
$724$	23.1194	0.859227
$725$	0	0
$726$	0	0
$727$	11.9083i	0.441655i	0.975313	+	0.220828i	$0.0708758\pi$
$727$	11.9083i	0.441655i	−0.975313	+	0.220828i	$0.929124\pi$
$728$	3.90833i	0.144852i
$729$	0	0
$730$	0	0
$731$	13.1194	0.485240
$732$	0	0
$733$	24.7250i	0.913238i	0.889662	+	0.456619i	$0.150940\pi$
$733$	24.7250i	0.913238i	−0.889662	+	0.456619i	$0.849060\pi$
$734$	−30.6333	−1.13070
$735$	0	0
$736$	36.6333	1.35032
$737$	− 29.7250i	− 1.09493i
$738$	0	0
$739$	−33.3305	−1.22608	−0.613042	−	0.790051i	$-0.710054\pi$
$739$	−33.3305	−1.22608	−0.613042	+	0.790051i	$0.710054\pi$
$740$	0	0
$741$	0	0
$742$	− 54.6333i	− 2.00565i
$743$	6.97224i	0.255787i	0.991788	+	0.127893i	$0.0408215\pi$
$743$	6.97224i	0.255787i	−0.991788	+	0.127893i	$0.959178\pi$
$744$	0	0
$745$	0	0
$746$	−19.3305	−0.707741
$747$	0	0
$748$	− 40.3305i	− 1.47463i
$749$	38.7250	1.41498
$750$	0	0
$751$	34.5139	1.25943	0.629715	−	0.776827i	$-0.283172\pi$
$751$	34.5139	1.25943	0.629715	+	0.776827i	$0.283172\pi$
$752$	1.88057i	0.0685774i
$753$	0	0
$754$	−0.972244	−0.0354070
$755$	0	0
$756$	0	0
$757$	− 20.1194i	− 0.731253i	−0.930762	−	0.365627i	$-0.880855\pi$
$757$	− 20.1194i	− 0.731253i	0.930762	−	0.365627i	$-0.119145\pi$
$758$	− 13.1194i	− 0.476519i
$759$	0	0
$760$	0	0
$761$	36.3583	1.31799	0.658993	−	0.752149i	$-0.270982\pi$
$761$	36.3583	1.31799	0.658993	+	0.752149i	$0.270982\pi$
$762$	0	0
$763$	− 51.2389i	− 1.85497i
$764$	30.4222	1.10064
$765$	0	0
$766$	13.1833	0.476334
$767$	1.11943i	0.0404203i
$768$	0	0
$769$	25.3583	0.914443	0.457222	−	0.889353i	$-0.348845\pi$
$769$	25.3583	0.914443	0.457222	+	0.889353i	$0.348845\pi$
$770$	0	0
$771$	0	0
$772$	22.5139i	0.810292i
$773$	− 44.4500i	− 1.59875i	−0.600830	−	0.799377i	$-0.705163\pi$
$773$	− 44.4500i	− 1.59875i	0.600830	−	0.799377i	$-0.294837\pi$
$774$	0	0
$775$	0	0
$776$	6.00000	0.215387
$777$	0	0
$778$	2.72498i	0.0976953i
$779$	−38.2389	−1.37005
$780$	0	0
$781$	−23.3028	−0.833839
$782$	− 36.6333i	− 1.31000i
$783$	0	0
$784$	−3.48612	−0.124504
$785$	0	0
$786$	0	0
$787$	46.5139i	1.65804i	0.559218	+	0.829020i	$0.311101\pi$
$787$	46.5139i	1.65804i	−0.559218	+	0.829020i	$0.688899\pi$
$788$	6.90833i	0.246099i
$789$	0	0
$790$	0	0
$791$	48.6333	1.72920
$792$	0	0
$793$	− 0.0916731i	− 0.00325541i
$794$	33.1472	1.17635
$795$	0	0
$796$	−24.7250	−0.876354
$797$	− 19.1194i	− 0.677245i	−0.940922	−	0.338622i	$-0.890039\pi$
$797$	− 19.1194i	− 0.677245i	0.940922	−	0.338622i	$-0.109961\pi$
$798$	0	0
$799$	−14.3028	−0.505996
$800$	0	0
$801$	0	0
$802$	9.63331i	0.340164i
$803$	− 50.9361i	− 1.79750i
$804$	0	0
$805$	0	0
$806$	−5.72498	−0.201654
$807$	0	0
$808$	44.1749i	1.55407i
$809$	32.0278	1.12604	0.563018	−	0.826445i	$-0.309640\pi$
$809$	32.0278	1.12604	0.563018	+	0.826445i	$0.309640\pi$
$810$	0	0
$811$	16.9361	0.594706	0.297353	−	0.954768i	$-0.403896\pi$
$811$	16.9361	0.594706	0.297353	+	0.954768i	$0.403896\pi$
$812$	− 19.8167i	− 0.695428i
$813$	0	0
$814$	23.3028	0.816762
$815$	0	0
$816$	0	0
$817$	− 41.0833i	− 1.43732i
$818$	− 70.3305i	− 2.45905i
$819$	0	0
$820$	0	0
$821$	33.0000	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$821$	33.0000	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$822$	0	0
$823$	− 50.6333i	− 1.76497i	−0.470344	−	0.882483i	$-0.655870\pi$
$823$	− 50.6333i	− 1.76497i	0.470344	−	0.882483i	$-0.344130\pi$
$824$	−53.7250	−1.87160
$825$	0	0
$826$	−36.6333	−1.27464
$827$	− 49.9638i	− 1.73741i	−0.495327	−	0.868706i	$-0.664952\pi$
$827$	− 49.9638i	− 1.73741i	0.495327	−	0.868706i	$-0.335048\pi$
$828$	0	0
$829$	12.7889	0.444177	0.222088	−	0.975027i	$-0.428713\pi$
$829$	12.7889	0.444177	0.222088	+	0.975027i	$0.428713\pi$
$830$	0	0
$831$	0	0
$832$	3.88057i	0.134535i
$833$	− 26.5139i	− 0.918651i
$834$	0	0
$835$	0	0
$836$	−126.294	−4.36798
$837$	0	0
$838$	81.2944i	2.80827i
$839$	47.0278	1.62358	0.811789	−	0.583951i	$-0.198494\pi$
$839$	47.0278	1.62358	0.811789	+	0.583951i	$0.198494\pi$
$840$	0	0
$841$	−27.0555	−0.932949
$842$	29.0278i	1.00036i
$843$	0	0
$844$	−16.5139	−0.568431
$845$	0	0
$846$	0	0
$847$	73.6611i	2.53102i
$848$	1.66947i	0.0573298i
$849$	0	0
$850$	0	0
$851$	13.1833	0.451919
$852$	0	0
$853$	− 43.6611i	− 1.49493i	−0.664303	−	0.747463i	$-0.731272\pi$
$853$	− 43.6611i	− 1.49493i	0.664303	−	0.747463i	$-0.268728\pi$
$854$	3.00000	0.102658
$855$	0	0
$856$	−27.0000	−0.922841
$857$	− 42.1472i	− 1.43972i	−0.694119	−	0.719860i	$-0.744206\pi$
$857$	− 42.1472i	− 1.43972i	0.694119	−	0.719860i	$-0.255794\pi$
$858$	0	0
$859$	27.6611	0.943783	0.471892	−	0.881657i	$-0.343571\pi$
$859$	27.6611	0.943783	0.471892	+	0.881657i	$0.343571\pi$
$860$	0	0
$861$	0	0
$862$	− 24.9083i	− 0.848381i
$863$	22.7527i	0.774512i	0.921972	+	0.387256i	$0.126577\pi$
$863$	22.7527i	0.774512i	−0.921972	+	0.387256i	$0.873423\pi$
$864$	0	0
$865$	0	0
$866$	37.3944	1.27072
$867$	0	0
$868$	− 116.689i	− 3.96068i
$869$	13.8167	0.468698
$870$	0	0
$871$	1.69722	0.0575083
$872$	35.7250i	1.20980i
$873$	0	0
$874$	−114.717	−3.88035
$875$	0	0
$876$	0	0
$877$	24.1194i	0.814455i	0.913327	+	0.407228i	$0.133504\pi$
$877$	24.1194i	0.814455i	−0.913327	+	0.407228i	$0.866496\pi$
$878$	31.0555i	1.04807i
$879$	0	0
$880$	0	0
$881$	−31.6056	−1.06482	−0.532409	−	0.846487i	$-0.678713\pi$
$881$	−31.6056	−1.06482	−0.532409	+	0.846487i	$0.678713\pi$
$882$	0	0
$883$	35.8167i	1.20533i	0.797996	+	0.602663i	$0.205894\pi$
$883$	35.8167i	1.20533i	−0.797996	+	0.602663i	$0.794106\pi$
$884$	2.30278	0.0774507
$885$	0	0
$886$	66.9916	2.25063
$887$	− 31.5416i	− 1.05906i	−0.848290	−	0.529532i	$-0.822367\pi$
$887$	− 31.5416i	− 1.05906i	0.848290	−	0.529532i	$-0.177633\pi$
$888$	0	0
$889$	−15.5139	−0.520319
$890$	0	0
$891$	0	0
$892$	56.1472i	1.87995i
$893$	44.7889i	1.49880i
$894$	0	0
$895$	0	0
$896$	−81.3583	−2.71799
$897$	0	0
$898$	4.33053i	0.144512i
$899$	11.4500	0.381878
$900$	0	0
$901$	−12.6972	−0.423006
$902$	− 64.7527i	− 2.15603i
$903$	0	0
$904$	−33.9083	−1.12777
$905$	0	0
$906$	0	0
$907$	27.7527i	0.921515i	0.887526	+	0.460757i	$0.152422\pi$
$907$	27.7527i	0.921515i	−0.887526	+	0.460757i	$0.847578\pi$
$908$	− 84.5694i	− 2.80653i
$909$	0	0
$910$	0	0
$911$	−45.4222	−1.50490	−0.752452	−	0.658647i	$-0.771129\pi$
$911$	−45.4222	−1.50490	−0.752452	+	0.658647i	$0.771129\pi$
$912$	0	0
$913$	47.7250i	1.57947i
$914$	8.23886	0.272517
$915$	0	0
$916$	−64.1472	−2.11948
$917$	− 7.81665i	− 0.258129i
$918$	0	0
$919$	−39.3305	−1.29739	−0.648697	−	0.761047i	$-0.724686\pi$
$919$	−39.3305	−1.29739	−0.648697	+	0.761047i	$0.724686\pi$
$920$	0	0
$921$	0	0
$922$	54.6333i	1.79925i
$923$	− 1.33053i	− 0.0437950i
$924$	0	0
$925$	0	0
$926$	−39.9083	−1.31147
$927$	0	0
$928$	− 7.39445i	− 0.242735i
$929$	−56.8722	−1.86592	−0.932958	−	0.359986i	$-0.882781\pi$
$929$	−56.8722	−1.86592	−0.932958	+	0.359986i	$0.882781\pi$
$930$	0	0
$931$	−83.0278	−2.72112
$932$	− 23.7250i	− 0.777138i
$933$	0	0
$934$	12.8444	0.420282
$935$	0	0
$936$	0	0
$937$	− 49.4222i	− 1.61455i	−0.590173	−	0.807277i	$-0.700941\pi$
$937$	− 49.4222i	− 1.61455i	0.590173	−	0.807277i	$-0.299059\pi$
$938$	55.5416i	1.81350i
$939$	0	0
$940$	0	0
$941$	−18.4861	−0.602630	−0.301315	−	0.953525i	$-0.597426\pi$
$941$	−18.4861	−0.602630	−0.301315	+	0.953525i	$0.597426\pi$
$942$	0	0
$943$	− 36.6333i	− 1.19294i
$944$	1.11943	0.0364343
$945$	0	0
$946$	69.5694	2.26190
$947$	34.8167i	1.13139i	0.824615	+	0.565695i	$0.191392\pi$
$947$	34.8167i	1.13139i	−0.824615	+	0.565695i	$0.808608\pi$
$948$	0	0
$949$	2.90833	0.0944083
$950$	0	0
$951$	0	0
$952$	29.7250i	0.963392i
$953$	12.4222i	0.402395i	0.979551	+	0.201197i	$0.0644833\pi$
$953$	12.4222i	0.402395i	−0.979551	+	0.201197i	$0.935517\pi$
$954$	0	0
$955$	0	0
$956$	−80.6611	−2.60877
$957$	0	0
$958$	− 78.0833i	− 2.52276i
$959$	55.5416	1.79353
$960$	0	0
$961$	36.4222	1.17491
$962$	1.33053i	0.0428981i
$963$	0	0
$964$	−15.8167	−0.509420
$965$	0	0
$966$	0	0
$967$	− 2.81665i	− 0.0905775i	−0.998974	−	0.0452887i	$-0.985579\pi$
$967$	− 2.81665i	− 0.0905775i	0.998974	−	0.0452887i	$-0.0144208\pi$
$968$	− 51.3583i	− 1.65072i
$969$	0	0
$970$	0	0
$971$	10.5416	0.338297	0.169149	−	0.985591i	$-0.445898\pi$
$971$	10.5416	0.338297	0.169149	+	0.985591i	$0.445898\pi$
$972$	0	0
$973$	34.0278i	1.09088i
$974$	17.5778	0.563229
$975$	0	0
$976$	−0.0916731	−0.00293438
$977$	16.1194i	0.515706i	0.966184	+	0.257853i	$0.0830151\pi$
$977$	16.1194i	0.515706i	−0.966184	+	0.257853i	$0.916985\pi$
$978$	0	0
$979$	−63.6333	−2.03373
$980$	0	0
$981$	0	0
$982$	53.5139i	1.70770i
$983$	− 12.6333i	− 0.402940i	−0.979495	−	0.201470i	$-0.935428\pi$
$983$	− 12.6333i	− 0.402940i	0.979495	−	0.201470i	$-0.0645718\pi$
$984$	0	0
$985$	0	0
$986$	−7.39445	−0.235487
$987$	0	0
$988$	− 7.21110i	− 0.229416i
$989$	39.3583	1.25152
$990$	0	0
$991$	−21.0917	−0.669999	−0.335000	−	0.942218i	$-0.608736\pi$
$991$	−21.0917	−0.669999	−0.335000	+	0.942218i	$0.608736\pi$
$992$	− 43.5416i	− 1.38245i
$993$	0	0
$994$	43.5416	1.38106
$995$	0	0
$996$	0	0
$997$	− 30.2389i	− 0.957674i	−0.877904	−	0.478837i	$-0.841058\pi$
$997$	− 30.2389i	− 0.957674i	0.877904	−	0.478837i	$-0.158942\pi$
$998$	− 76.7527i	− 2.42957i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.2.b.j.649.4		4
3.2	odd	2		2025.2.b.i.649.1		4
5.2	odd	4		2025.2.a.h.1.1	✓	2
5.3	odd	4		2025.2.a.l.1.2	yes	2
5.4	even	2	inner	2025.2.b.j.649.1		4
15.2	even	4		2025.2.a.k.1.2	yes	2
15.8	even	4		2025.2.a.i.1.1	yes	2
15.14	odd	2		2025.2.b.i.649.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2025.2.a.h.1.1	✓	2	5.2	odd	4
2025.2.a.i.1.1	yes	2	15.8	even	4
2025.2.a.k.1.2	yes	2	15.2	even	4
2025.2.a.l.1.2	yes	2	5.3	odd	4
2025.2.b.i.649.1		4	3.2	odd	2
2025.2.b.i.649.4		4	15.14	odd	2
2025.2.b.j.649.1		4	5.4	even	2	inner
2025.2.b.j.649.4		4	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.30278i q^{2} -3.30278 q^{4} +4.30278i q^{7} -3.00000i q^{8} +O(q^{10})\)	\(q+2.30278i q^{2} -3.30278 q^{4} +4.30278i q^{7} -3.00000i q^{8} +5.30278 q^{11} +0.302776i q^{13} -9.90833 q^{14} +0.302776 q^{16} +2.30278i q^{17} +7.21110 q^{19} +12.2111i q^{22} +6.90833i q^{23} -0.697224 q^{26} -14.2111i q^{28} +1.39445 q^{29} +8.21110 q^{31} -5.30278i q^{32} -5.30278 q^{34} -1.90833i q^{37} +16.6056i q^{38} -5.30278 q^{41} -5.69722i q^{43} -17.5139 q^{44} -15.9083 q^{46} +6.21110i q^{47} -11.5139 q^{49} -1.00000i q^{52} +5.51388i q^{53} +12.9083 q^{56} +3.21110i q^{58} +3.69722 q^{59} -0.302776 q^{61} +18.9083i q^{62} +12.8167 q^{64} -5.60555i q^{67} -7.60555i q^{68} -4.39445 q^{71} -9.60555i q^{73} +4.39445 q^{74} -23.8167 q^{76} +22.8167i q^{77} +2.60555 q^{79} -12.2111i q^{82} +9.00000i q^{83} +13.1194 q^{86} -15.9083i q^{88} -12.0000 q^{89} -1.30278 q^{91} -22.8167i q^{92} -14.3028 q^{94} +2.00000i q^{97} -26.5139i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{4}+O(q^{10}) \) 4 * q - 6 * q^4	\( 4 q - 6 q^{4} + 14 q^{11} - 18 q^{14} - 6 q^{16} - 10 q^{26} + 20 q^{29} + 4 q^{31} - 14 q^{34} - 14 q^{41} - 34 q^{44} - 42 q^{46} - 10 q^{49} + 30 q^{56} + 22 q^{59} + 6 q^{61} + 8 q^{64} - 32 q^{71} + 32 q^{74} - 52 q^{76} - 4 q^{79} + 2 q^{86} - 48 q^{89} + 2 q^{91} - 50 q^{94}+O(q^{100}) \) 4 * q - 6 * q^4 + 14 * q^11 - 18 * q^14 - 6 * q^16 - 10 * q^26 + 20 * q^29 + 4 * q^31 - 14 * q^34 - 14 * q^41 - 34 * q^44 - 42 * q^46 - 10 * q^49 + 30 * q^56 + 22 * q^59 + 6 * q^61 + 8 * q^64 - 32 * q^71 + 32 * q^74 - 52 * q^76 - 4 * q^79 + 2 * q^86 - 48 * q^89 + 2 * q^91 - 50 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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