LMFDB - Embedded newform 2475.2.c.o.199.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(199,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$19.7629745003$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 825)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2475.199
Dual form			2475.2.c.o.199.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.41421i q^{2} -3.82843 q^{4} +0.414214i q^{7} +4.41421i q^{8} +O(q^{10})$	$q-2.41421i q^{2} -3.82843 q^{4} +0.414214i q^{7} +4.41421i q^{8} +1.00000 q^{11} +2.82843i q^{13} +1.00000 q^{14} +3.00000 q^{16} -2.41421i q^{17} -6.41421 q^{19} -2.41421i q^{22} +1.00000i q^{23} +6.82843 q^{26} -1.58579i q^{28} +1.17157 q^{29} -8.48528 q^{31} +1.58579i q^{32} -5.82843 q^{34} -0.171573i q^{37} +15.4853i q^{38} +10.8995 q^{41} +11.6569i q^{43} -3.82843 q^{44} +2.41421 q^{46} +7.48528i q^{47} +6.82843 q^{49} -10.8284i q^{52} +7.65685i q^{53} -1.82843 q^{56} -2.82843i q^{58} +11.0000 q^{59} +8.82843 q^{61} +20.4853i q^{62} +9.82843 q^{64} -0.343146i q^{67} +9.24264i q^{68} -7.82843 q^{71} +8.82843i q^{73} -0.414214 q^{74} +24.5563 q^{76} +0.414214i q^{77} -13.2426 q^{79} -26.3137i q^{82} -4.48528i q^{83} +28.1421 q^{86} +4.41421i q^{88} +3.65685 q^{89} -1.17157 q^{91} -3.82843i q^{92} +18.0711 q^{94} +5.82843i q^{97} -16.4853i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{11} + 4 q^{14} + 12 q^{16} - 20 q^{19} + 16 q^{26} + 16 q^{29} - 12 q^{34} + 4 q^{41} - 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{56} + 44 q^{59} + 24 q^{61} + 28 q^{64} - 20 q^{71} + 4 q^{74} + 36 q^{76} - 36 q^{79} + 56 q^{86} - 8 q^{89} - 16 q^{91} + 44 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^11 + 4 * q^14 + 12 * q^16 - 20 * q^19 + 16 * q^26 + 16 * q^29 - 12 * q^34 + 4 * q^41 - 4 * q^44 + 4 * q^46 + 16 * q^49 + 4 * q^56 + 44 * q^59 + 24 * q^61 + 28 * q^64 - 20 * q^71 + 4 * q^74 + 36 * q^76 - 36 * q^79 + 56 * q^86 - 8 * q^89 - 16 * q^91 + 44 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times$.

$n$	$551$	$2026$	$2377$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$	− 2.41421i	− 1.70711i	−0.521005	−	0.853553i	$-0.674443\pi$
$2$	− 2.41421i	− 1.70711i	0.521005	−	0.853553i	$-0.325557\pi$
$3$	0	0
$3$	0	0
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.414214i	0.156558i	0.996931	+	0.0782790i	$0.0249425\pi$
$7$	0.414214i	0.156558i	−0.996931	+	0.0782790i	$0.975058\pi$
$8$	4.41421i	1.56066i
$9$	0	0
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.82843i	0.784465i	0.919866	+	0.392232i	$0.128297\pi$
$13$	2.82843i	0.784465i	−0.919866	+	0.392232i	$0.871703\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	3.00000	0.750000
$17$	− 2.41421i	− 0.585533i	−0.956184	−	0.292766i	$-0.905424\pi$
$17$	− 2.41421i	− 0.585533i	0.956184	−	0.292766i	$-0.0945758\pi$
$18$	0	0
$19$	−6.41421	−1.47152	−0.735761	−	0.677242i	$-0.763175\pi$
$19$	−6.41421	−1.47152	−0.735761	+	0.677242i	$0.763175\pi$
$20$	0	0
$21$	0	0
$22$	− 2.41421i	− 0.514712i
$23$	1.00000i	0.208514i	0.994550	+	0.104257i	$0.0332465\pi$
$23$	1.00000i	0.208514i	−0.994550	+	0.104257i	$0.966753\pi$
$24$	0	0
$25$	0	0
$26$	6.82843	1.33916
$27$	0	0
$28$	− 1.58579i	− 0.299685i
$29$	1.17157	0.217556	0.108778	−	0.994066i	$-0.465306\pi$
$29$	1.17157	0.217556	0.108778	+	0.994066i	$0.465306\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	1.58579i	0.280330i
$33$	0	0
$34$	−5.82843	−0.999567
$35$	0	0
$36$	0	0
$37$	− 0.171573i	− 0.0282064i	−0.999901	−	0.0141032i	$-0.995511\pi$
$37$	− 0.171573i	− 0.0282064i	0.999901	−	0.0141032i	$-0.00448934\pi$
$38$	15.4853i	2.51204i
$39$	0	0
$40$	0	0
$41$	10.8995	1.70222	0.851108	−	0.524991i	$-0.175931\pi$
$41$	10.8995	1.70222	0.851108	+	0.524991i	$0.175931\pi$
$42$	0	0
$43$	11.6569i	1.77765i	0.458243	+	0.888827i	$0.348479\pi$
$43$	11.6569i	1.77765i	−0.458243	+	0.888827i	$0.651521\pi$
$44$	−3.82843	−0.577157
$45$	0	0
$46$	2.41421	0.355956
$47$	7.48528i	1.09184i	0.837837	+	0.545920i	$0.183820\pi$
$47$	7.48528i	1.09184i	−0.837837	+	0.545920i	$0.816180\pi$
$48$	0	0
$49$	6.82843	0.975490
$50$	0	0
$51$	0	0
$52$	− 10.8284i	− 1.50163i
$53$	7.65685i	1.05175i	0.850562	+	0.525875i	$0.176262\pi$
$53$	7.65685i	1.05175i	−0.850562	+	0.525875i	$0.823738\pi$
$54$	0	0
$55$	0	0
$56$	−1.82843	−0.244334
$57$	0	0
$58$	− 2.82843i	− 0.371391i
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	0	0
$61$	8.82843	1.13036	0.565182	−	0.824966i	$-0.308806\pi$
$61$	8.82843	1.13036	0.565182	+	0.824966i	$0.308806\pi$
$62$	20.4853i	2.60163i
$63$	0	0
$64$	9.82843	1.22855
$65$	0	0
$66$	0	0
$67$	− 0.343146i	− 0.0419219i	−0.999780	−	0.0209610i	$-0.993327\pi$
$67$	− 0.343146i	− 0.0419219i	0.999780	−	0.0209610i	$-0.00667257\pi$
$68$	9.24264i	1.12083i
$69$	0	0
$70$	0	0
$71$	−7.82843	−0.929063	−0.464532	−	0.885556i	$-0.653777\pi$
$71$	−7.82843	−0.929063	−0.464532	+	0.885556i	$0.653777\pi$
$72$	0	0
$73$	8.82843i	1.03329i	0.856200	+	0.516645i	$0.172819\pi$
$73$	8.82843i	1.03329i	−0.856200	+	0.516645i	$0.827181\pi$
$74$	−0.414214	−0.0481513
$75$	0	0
$76$	24.5563	2.81681
$77$	0.414214i	0.0472040i
$78$	0	0
$79$	−13.2426	−1.48991	−0.744957	−	0.667113i	$-0.767530\pi$
$79$	−13.2426	−1.48991	−0.744957	+	0.667113i	$0.767530\pi$
$80$	0	0
$81$	0	0
$82$	− 26.3137i	− 2.90586i
$83$	− 4.48528i	− 0.492324i	−0.969229	−	0.246162i	$-0.920831\pi$
$83$	− 4.48528i	− 0.492324i	0.969229	−	0.246162i	$-0.0791695\pi$
$84$	0	0
$85$	0	0
$86$	28.1421	3.03464
$87$	0	0
$88$	4.41421i	0.470557i
$89$	3.65685	0.387626	0.193813	−	0.981039i	$-0.437915\pi$
$89$	3.65685	0.387626	0.193813	+	0.981039i	$0.437915\pi$
$90$	0	0
$91$	−1.17157	−0.122814
$92$	− 3.82843i	− 0.399141i
$93$	0	0
$94$	18.0711	1.86389
$95$	0	0
$96$	0	0
$97$	5.82843i	0.591787i	0.955221	+	0.295894i	$0.0956174\pi$
$97$	5.82843i	0.591787i	−0.955221	+	0.295894i	$0.904383\pi$
$98$	− 16.4853i	− 1.66526i
$99$	0	0
$100$	0	0
$101$	−14.8995	−1.48256	−0.741278	−	0.671199i	$-0.765780\pi$
$101$	−14.8995	−1.48256	−0.741278	+	0.671199i	$0.765780\pi$
$102$	0	0
$103$	13.6569i	1.34565i	0.739802	+	0.672825i	$0.234919\pi$
$103$	13.6569i	1.34565i	−0.739802	+	0.672825i	$0.765081\pi$
$104$	−12.4853	−1.22428
$105$	0	0
$106$	18.4853	1.79545
$107$	5.31371i	0.513696i	0.966452	+	0.256848i	$0.0826839\pi$
$107$	5.31371i	0.513696i	−0.966452	+	0.256848i	$0.917316\pi$
$108$	0	0
$109$	−5.31371	−0.508961	−0.254480	−	0.967078i	$-0.581904\pi$
$109$	−5.31371	−0.508961	−0.254480	+	0.967078i	$0.581904\pi$
$110$	0	0
$111$	0	0
$112$	1.24264i	0.117419i
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	0	0
$115$	0	0
$116$	−4.48528	−0.416448
$117$	0	0
$118$	− 26.5563i	− 2.44471i
$119$	1.00000	0.0916698
$120$	0	0
$121$	1.00000	0.0909091
$122$	− 21.3137i	− 1.92965i
$123$	0	0
$124$	32.4853	2.91726
$125$	0	0
$126$	0	0
$127$	7.24264i	0.642680i	0.946964	+	0.321340i	$0.104133\pi$
$127$	7.24264i	0.642680i	−0.946964	+	0.321340i	$0.895867\pi$
$128$	− 20.5563i	− 1.81694i
$129$	0	0
$130$	0	0
$131$	−6.82843	−0.596602	−0.298301	−	0.954472i	$-0.596420\pi$
$131$	−6.82843	−0.596602	−0.298301	+	0.954472i	$0.596420\pi$
$132$	0	0
$133$	− 2.65685i	− 0.230378i
$134$	−0.828427	−0.0715652
$135$	0	0
$136$	10.6569	0.913818
$137$	12.1421i	1.03737i	0.854965	+	0.518686i	$0.173579\pi$
$137$	12.1421i	1.03737i	−0.854965	+	0.518686i	$0.826421\pi$
$138$	0	0
$139$	18.9706	1.60906	0.804531	−	0.593911i	$-0.202417\pi$
$139$	18.9706	1.60906	0.804531	+	0.593911i	$0.202417\pi$
$140$	0	0
$141$	0	0
$142$	18.8995i	1.58601i
$143$	2.82843i	0.236525i
$144$	0	0
$145$	0	0
$146$	21.3137	1.76394
$147$	0	0
$148$	0.656854i	0.0539931i
$149$	7.72792	0.633096	0.316548	−	0.948576i	$-0.397476\pi$
$149$	7.72792	0.633096	0.316548	+	0.948576i	$0.397476\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	− 28.3137i	− 2.29655i
$153$	0	0
$154$	1.00000	0.0805823
$155$	0	0
$156$	0	0
$157$	6.00000i	0.478852i	0.970915	+	0.239426i	$0.0769593\pi$
$157$	6.00000i	0.478852i	−0.970915	+	0.239426i	$0.923041\pi$
$158$	31.9706i	2.54344i
$159$	0	0
$160$	0	0
$161$	−0.414214	−0.0326446
$162$	0	0
$163$	− 15.7990i	− 1.23747i	−0.785599	−	0.618736i	$-0.787645\pi$
$163$	− 15.7990i	− 1.23747i	0.785599	−	0.618736i	$-0.212355\pi$
$164$	−41.7279	−3.25840
$165$	0	0
$166$	−10.8284	−0.840449
$167$	− 21.7990i	− 1.68686i	−0.537242	−	0.843428i	$-0.680534\pi$
$167$	− 21.7990i	− 1.68686i	0.537242	−	0.843428i	$-0.319466\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	− 44.6274i	− 3.40281i
$173$	− 12.5563i	− 0.954642i	−0.878729	−	0.477321i	$-0.841608\pi$
$173$	− 12.5563i	− 0.954642i	0.878729	−	0.477321i	$-0.158392\pi$
$174$	0	0
$175$	0	0
$176$	3.00000	0.226134
$177$	0	0
$178$	− 8.82843i	− 0.661719i
$179$	16.7990	1.25562	0.627808	−	0.778368i	$-0.283952\pi$
$179$	16.7990	1.25562	0.627808	+	0.778368i	$0.283952\pi$
$180$	0	0
$181$	−21.9706	−1.63306	−0.816530	−	0.577304i	$-0.804105\pi$
$181$	−21.9706	−1.63306	−0.816530	+	0.577304i	$0.804105\pi$
$182$	2.82843i	0.209657i
$183$	0	0
$184$	−4.41421	−0.325420
$185$	0	0
$186$	0	0
$187$	− 2.41421i	− 0.176545i
$188$	− 28.6569i	− 2.09002i
$189$	0	0
$190$	0	0
$191$	−6.17157	−0.446559	−0.223280	−	0.974754i	$-0.571676\pi$
$191$	−6.17157	−0.446559	−0.223280	+	0.974754i	$0.571676\pi$
$192$	0	0
$193$	− 3.31371i	− 0.238526i	−0.992863	−	0.119263i	$-0.961947\pi$
$193$	− 3.31371i	− 0.238526i	0.992863	−	0.119263i	$-0.0380532\pi$
$194$	14.0711	1.01024
$195$	0	0
$196$	−26.1421	−1.86730
$197$	− 4.75736i	− 0.338948i	−0.985535	−	0.169474i	$-0.945793\pi$
$197$	− 4.75736i	− 0.338948i	0.985535	−	0.169474i	$-0.0542068\pi$
$198$	0	0
$199$	−10.8284	−0.767607	−0.383803	−	0.923415i	$-0.625386\pi$
$199$	−10.8284	−0.767607	−0.383803	+	0.923415i	$0.625386\pi$
$200$	0	0
$201$	0	0
$202$	35.9706i	2.53088i
$203$	0.485281i	0.0340601i
$204$	0	0
$205$	0	0
$206$	32.9706	2.29717
$207$	0	0
$208$	8.48528i	0.588348i
$209$	−6.41421	−0.443680
$210$	0	0
$211$	−13.3137	−0.916553	−0.458277	−	0.888810i	$-0.651533\pi$
$211$	−13.3137	−0.916553	−0.458277	+	0.888810i	$0.651533\pi$
$212$	− 29.3137i	− 2.01327i
$213$	0	0
$214$	12.8284	0.876933
$215$	0	0
$216$	0	0
$217$	− 3.51472i	− 0.238595i
$218$	12.8284i	0.868851i
$219$	0	0
$220$	0	0
$221$	6.82843	0.459330
$222$	0	0
$223$	21.1716i	1.41775i	0.705332	+	0.708877i	$0.250798\pi$
$223$	21.1716i	1.41775i	−0.705332	+	0.708877i	$0.749202\pi$
$224$	−0.656854	−0.0438879
$225$	0	0
$226$	−24.1421	−1.60591
$227$	18.4853i	1.22691i	0.789729	+	0.613456i	$0.210221\pi$
$227$	18.4853i	1.22691i	−0.789729	+	0.613456i	$0.789779\pi$
$228$	0	0
$229$	2.51472	0.166177	0.0830886	−	0.996542i	$-0.473522\pi$
$229$	2.51472	0.166177	0.0830886	+	0.996542i	$0.473522\pi$
$230$	0	0
$231$	0	0
$232$	5.17157i	0.339530i
$233$	16.5563i	1.08464i	0.840171	+	0.542321i	$0.182454\pi$
$233$	16.5563i	1.08464i	−0.840171	+	0.542321i	$0.817546\pi$
$234$	0	0
$235$	0	0
$236$	−42.1127	−2.74130
$237$	0	0
$238$	− 2.41421i	− 0.156490i
$239$	23.6569	1.53023	0.765117	−	0.643891i	$-0.222681\pi$
$239$	23.6569	1.53023	0.765117	+	0.643891i	$0.222681\pi$
$240$	0	0
$241$	14.1421	0.910975	0.455488	−	0.890242i	$-0.349465\pi$
$241$	14.1421	0.910975	0.455488	+	0.890242i	$0.349465\pi$
$242$	− 2.41421i	− 0.155192i
$243$	0	0
$244$	−33.7990	−2.16376
$245$	0	0
$246$	0	0
$247$	− 18.1421i	− 1.15436i
$248$	− 37.4558i	− 2.37845i
$249$	0	0
$250$	0	0
$251$	−8.97056	−0.566217	−0.283108	−	0.959088i	$-0.591366\pi$
$251$	−8.97056	−0.566217	−0.283108	+	0.959088i	$0.591366\pi$
$252$	0	0
$253$	1.00000i	0.0628695i
$254$	17.4853	1.09712
$255$	0	0
$256$	−29.9706	−1.87316
$257$	− 9.31371i	− 0.580973i	−0.956879	−	0.290487i	$-0.906183\pi$
$257$	− 9.31371i	− 0.580973i	0.956879	−	0.290487i	$-0.0938172\pi$
$258$	0	0
$259$	0.0710678	0.00441594
$260$	0	0
$261$	0	0
$262$	16.4853i	1.01846i
$263$	22.9706i	1.41643i	0.705999	+	0.708213i	$0.250498\pi$
$263$	22.9706i	1.41643i	−0.705999	+	0.708213i	$0.749502\pi$
$264$	0	0
$265$	0	0
$266$	−6.41421	−0.393281
$267$	0	0
$268$	1.31371i	0.0802475i
$269$	−15.7990	−0.963281	−0.481641	−	0.876369i	$-0.659959\pi$
$269$	−15.7990	−0.963281	−0.481641	+	0.876369i	$0.659959\pi$
$270$	0	0
$271$	8.89949	0.540606	0.270303	−	0.962775i	$-0.412876\pi$
$271$	8.89949	0.540606	0.270303	+	0.962775i	$0.412876\pi$
$272$	− 7.24264i	− 0.439150i
$273$	0	0
$274$	29.3137	1.77091
$275$	0	0
$276$	0	0
$277$	4.82843i	0.290112i	0.989423	+	0.145056i	$0.0463362\pi$
$277$	4.82843i	0.290112i	−0.989423	+	0.145056i	$0.953664\pi$
$278$	− 45.7990i	− 2.74684i
$279$	0	0
$280$	0	0
$281$	−32.0711	−1.91320	−0.956600	−	0.291405i	$-0.905877\pi$
$281$	−32.0711	−1.91320	−0.956600	+	0.291405i	$0.905877\pi$
$282$	0	0
$283$	0.899495i	0.0534694i	0.999643	+	0.0267347i	$0.00851094\pi$
$283$	0.899495i	0.0534694i	−0.999643	+	0.0267347i	$0.991489\pi$
$284$	29.9706	1.77843
$285$	0	0
$286$	6.82843	0.403773
$287$	4.51472i	0.266495i
$288$	0	0
$289$	11.1716	0.657151
$290$	0	0
$291$	0	0
$292$	− 33.7990i	− 1.97794i
$293$	17.5858i	1.02737i	0.857978	+	0.513686i	$0.171720\pi$
$293$	17.5858i	1.02737i	−0.857978	+	0.513686i	$0.828280\pi$
$294$	0	0
$295$	0	0
$296$	0.757359	0.0440206
$297$	0	0
$298$	− 18.6569i	− 1.08076i
$299$	−2.82843	−0.163572
$300$	0	0
$301$	−4.82843	−0.278306
$302$	− 33.7990i	− 1.94491i
$303$	0	0
$304$	−19.2426	−1.10364
$305$	0	0
$306$	0	0
$307$	6.68629i	0.381607i	0.981628	+	0.190803i	$0.0611093\pi$
$307$	6.68629i	0.381607i	−0.981628	+	0.190803i	$0.938891\pi$
$308$	− 1.58579i	− 0.0903586i
$309$	0	0
$310$	0	0
$311$	−13.6569	−0.774409	−0.387205	−	0.921994i	$-0.626559\pi$
$311$	−13.6569	−0.774409	−0.387205	+	0.921994i	$0.626559\pi$
$312$	0	0
$313$	27.1421i	1.53416i	0.641549	+	0.767082i	$0.278292\pi$
$313$	27.1421i	1.53416i	−0.641549	+	0.767082i	$0.721708\pi$
$314$	14.4853	0.817452
$315$	0	0
$316$	50.6985	2.85201
$317$	− 30.8284i	− 1.73150i	−0.500479	−	0.865748i	$-0.666843\pi$
$317$	− 30.8284i	− 1.73150i	0.500479	−	0.865748i	$-0.333157\pi$
$318$	0	0
$319$	1.17157	0.0655955
$320$	0	0
$321$	0	0
$322$	1.00000i	0.0557278i
$323$	15.4853i	0.861624i
$324$	0	0
$325$	0	0
$326$	−38.1421	−2.11250
$327$	0	0
$328$	48.1127i	2.65658i
$329$	−3.10051	−0.170936
$330$	0	0
$331$	−32.1421	−1.76669	−0.883346	−	0.468722i	$-0.844715\pi$
$331$	−32.1421	−1.76669	−0.883346	+	0.468722i	$0.844715\pi$
$332$	17.1716i	0.942412i
$333$	0	0
$334$	−52.6274	−2.87964
$335$	0	0
$336$	0	0
$337$	4.14214i	0.225637i	0.993616	+	0.112818i	$0.0359878\pi$
$337$	4.14214i	0.225637i	−0.993616	+	0.112818i	$0.964012\pi$
$338$	− 12.0711i	− 0.656580i
$339$	0	0
$340$	0	0
$341$	−8.48528	−0.459504
$342$	0	0
$343$	5.72792i	0.309279i
$344$	−51.4558	−2.77431
$345$	0	0
$346$	−30.3137	−1.62968
$347$	21.1716i	1.13655i	0.822839	+	0.568275i	$0.192389\pi$
$347$	21.1716i	1.13655i	−0.822839	+	0.568275i	$0.807611\pi$
$348$	0	0
$349$	2.48528	0.133034	0.0665170	−	0.997785i	$-0.478811\pi$
$349$	2.48528	0.133034	0.0665170	+	0.997785i	$0.478811\pi$
$350$	0	0
$351$	0	0
$352$	1.58579i	0.0845227i
$353$	− 4.48528i	− 0.238727i	−0.992851	−	0.119364i	$-0.961915\pi$
$353$	− 4.48528i	− 0.238727i	0.992851	−	0.119364i	$-0.0380855\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	− 40.5563i	− 2.14347i
$359$	15.5147	0.818836	0.409418	−	0.912347i	$-0.365732\pi$
$359$	15.5147	0.818836	0.409418	+	0.912347i	$0.365732\pi$
$360$	0	0
$361$	22.1421	1.16538
$362$	53.0416i	2.78781i
$363$	0	0
$364$	4.48528	0.235093
$365$	0	0
$366$	0	0
$367$	1.31371i	0.0685750i	0.999412	+	0.0342875i	$0.0109162\pi$
$367$	1.31371i	0.0685750i	−0.999412	+	0.0342875i	$0.989084\pi$
$368$	3.00000i	0.156386i
$369$	0	0
$370$	0	0
$371$	−3.17157	−0.164660
$372$	0	0
$373$	23.6569i	1.22491i	0.790507	+	0.612453i	$0.209817\pi$
$373$	23.6569i	1.22491i	−0.790507	+	0.612453i	$0.790183\pi$
$374$	−5.82843	−0.301381
$375$	0	0
$376$	−33.0416	−1.70399
$377$	3.31371i	0.170665i
$378$	0	0
$379$	−9.17157	−0.471112	−0.235556	−	0.971861i	$-0.575691\pi$
$379$	−9.17157	−0.471112	−0.235556	+	0.971861i	$0.575691\pi$
$380$	0	0
$381$	0	0
$382$	14.8995i	0.762324i
$383$	20.0000i	1.02195i	0.859595	+	0.510976i	$0.170716\pi$
$383$	20.0000i	1.02195i	−0.859595	+	0.510976i	$0.829284\pi$
$384$	0	0
$385$	0	0
$386$	−8.00000	−0.407189
$387$	0	0
$388$	− 22.3137i	− 1.13281i
$389$	−17.6569	−0.895238	−0.447619	−	0.894224i	$-0.647728\pi$
$389$	−17.6569	−0.895238	−0.447619	+	0.894224i	$0.647728\pi$
$390$	0	0
$391$	2.41421	0.122092
$392$	30.1421i	1.52241i
$393$	0	0
$394$	−11.4853	−0.578620
$395$	0	0
$396$	0	0
$397$	− 35.9411i	− 1.80383i	−0.431910	−	0.901917i	$-0.642160\pi$
$397$	− 35.9411i	− 1.80383i	0.431910	−	0.901917i	$-0.357840\pi$
$398$	26.1421i	1.31039i
$399$	0	0
$400$	0	0
$401$	−31.7990	−1.58797	−0.793983	−	0.607940i	$-0.791996\pi$
$401$	−31.7990	−1.58797	−0.793983	+	0.607940i	$0.791996\pi$
$402$	0	0
$403$	− 24.0000i	− 1.19553i
$404$	57.0416	2.83793
$405$	0	0
$406$	1.17157	0.0581442
$407$	− 0.171573i	− 0.00850455i
$408$	0	0
$409$	4.14214	0.204815	0.102408	−	0.994743i	$-0.467345\pi$
$409$	4.14214	0.204815	0.102408	+	0.994743i	$0.467345\pi$
$410$	0	0
$411$	0	0
$412$	− 52.2843i	− 2.57586i
$413$	4.55635i	0.224203i
$414$	0	0
$415$	0	0
$416$	−4.48528	−0.219909
$417$	0	0
$418$	15.4853i	0.757410i
$419$	−25.4853	−1.24504	−0.622519	−	0.782605i	$-0.713891\pi$
$419$	−25.4853	−1.24504	−0.622519	+	0.782605i	$0.713891\pi$
$420$	0	0
$421$	−27.0000	−1.31590	−0.657950	−	0.753062i	$-0.728576\pi$
$421$	−27.0000	−1.31590	−0.657950	+	0.753062i	$0.728576\pi$
$422$	32.1421i	1.56465i
$423$	0	0
$424$	−33.7990	−1.64142
$425$	0	0
$426$	0	0
$427$	3.65685i	0.176968i
$428$	− 20.3431i	− 0.983323i
$429$	0	0
$430$	0	0
$431$	−1.17157	−0.0564327	−0.0282163	−	0.999602i	$-0.508983\pi$
$431$	−1.17157	−0.0564327	−0.0282163	+	0.999602i	$0.508983\pi$
$432$	0	0
$433$	13.3137i	0.639816i	0.947449	+	0.319908i	$0.103652\pi$
$433$	13.3137i	0.639816i	−0.947449	+	0.319908i	$0.896348\pi$
$434$	−8.48528	−0.407307
$435$	0	0
$436$	20.3431	0.974260
$437$	− 6.41421i	− 0.306833i
$438$	0	0
$439$	−2.27208	−0.108440	−0.0542202	−	0.998529i	$-0.517267\pi$
$439$	−2.27208	−0.108440	−0.0542202	+	0.998529i	$0.517267\pi$
$440$	0	0
$441$	0	0
$442$	− 16.4853i	− 0.784125i
$443$	− 7.97056i	− 0.378693i	−0.981910	−	0.189346i	$-0.939363\pi$
$443$	− 7.97056i	− 0.378693i	0.981910	−	0.189346i	$-0.0606369\pi$
$444$	0	0
$445$	0	0
$446$	51.1127	2.42026
$447$	0	0
$448$	4.07107i	0.192340i
$449$	−6.48528	−0.306059	−0.153030	−	0.988222i	$-0.548903\pi$
$449$	−6.48528	−0.306059	−0.153030	+	0.988222i	$0.548903\pi$
$450$	0	0
$451$	10.8995	0.513237
$452$	38.2843i	1.80074i
$453$	0	0
$454$	44.6274	2.09447
$455$	0	0
$456$	0	0
$457$	3.85786i	0.180463i	0.995921	+	0.0902316i	$0.0287607\pi$
$457$	3.85786i	0.180463i	−0.995921	+	0.0902316i	$0.971239\pi$
$458$	− 6.07107i	− 0.283682i
$459$	0	0
$460$	0	0
$461$	32.7696	1.52623	0.763115	−	0.646263i	$-0.223669\pi$
$461$	32.7696	1.52623	0.763115	+	0.646263i	$0.223669\pi$
$462$	0	0
$463$	− 34.9706i	− 1.62522i	−0.582808	−	0.812610i	$-0.698046\pi$
$463$	− 34.9706i	− 1.62522i	0.582808	−	0.812610i	$-0.301954\pi$
$464$	3.51472	0.163167
$465$	0	0
$466$	39.9706	1.85160
$467$	− 10.6274i	− 0.491778i	−0.969298	−	0.245889i	$-0.920920\pi$
$467$	− 10.6274i	− 0.491778i	0.969298	−	0.245889i	$-0.0790799\pi$
$468$	0	0
$469$	0.142136	0.00656321
$470$	0	0
$471$	0	0
$472$	48.5563i	2.23499i
$473$	11.6569i	0.535983i
$474$	0	0
$475$	0	0
$476$	−3.82843	−0.175476
$477$	0	0
$478$	− 57.1127i	− 2.61227i
$479$	−24.4853	−1.11876	−0.559381	−	0.828911i	$-0.688961\pi$
$479$	−24.4853	−1.11876	−0.559381	+	0.828911i	$0.688961\pi$
$480$	0	0
$481$	0.485281	0.0221269
$482$	− 34.1421i	− 1.55513i
$483$	0	0
$484$	−3.82843	−0.174019
$485$	0	0
$486$	0	0
$487$	− 6.48528i	− 0.293876i	−0.989146	−	0.146938i	$-0.953058\pi$
$487$	− 6.48528i	− 0.293876i	0.989146	−	0.146938i	$-0.0469418\pi$
$488$	38.9706i	1.76411i
$489$	0	0
$490$	0	0
$491$	24.1421	1.08952	0.544760	−	0.838592i	$-0.316621\pi$
$491$	24.1421	1.08952	0.544760	+	0.838592i	$0.316621\pi$
$492$	0	0
$493$	− 2.82843i	− 0.127386i
$494$	−43.7990	−1.97061
$495$	0	0
$496$	−25.4558	−1.14300
$497$	− 3.24264i	− 0.145452i
$498$	0	0
$499$	−35.1716	−1.57450	−0.787248	−	0.616637i	$-0.788495\pi$
$499$	−35.1716	−1.57450	−0.787248	+	0.616637i	$0.788495\pi$
$500$	0	0
$501$	0	0
$502$	21.6569i	0.966593i
$503$	− 34.2843i	− 1.52866i	−0.644825	−	0.764330i	$-0.723070\pi$
$503$	− 34.2843i	− 1.52866i	0.644825	−	0.764330i	$-0.276930\pi$
$504$	0	0
$505$	0	0
$506$	2.41421	0.107325
$507$	0	0
$508$	− 27.7279i	− 1.23023i
$509$	4.62742	0.205107	0.102553	−	0.994728i	$-0.467299\pi$
$509$	4.62742	0.205107	0.102553	+	0.994728i	$0.467299\pi$
$510$	0	0
$511$	−3.65685	−0.161770
$512$	31.2426i	1.38074i
$513$	0	0
$514$	−22.4853	−0.991783
$515$	0	0
$516$	0	0
$517$	7.48528i	0.329202i
$518$	− 0.171573i	− 0.00753848i
$519$	0	0
$520$	0	0
$521$	36.1421	1.58342	0.791708	−	0.610900i	$-0.209192\pi$
$521$	36.1421	1.58342	0.791708	+	0.610900i	$0.209192\pi$
$522$	0	0
$523$	− 42.2132i	− 1.84585i	−0.384974	−	0.922927i	$-0.625790\pi$
$523$	− 42.2132i	− 1.84585i	0.384974	−	0.922927i	$-0.374210\pi$
$524$	26.1421	1.14202
$525$	0	0
$526$	55.4558	2.41799
$527$	20.4853i	0.892353i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	0	0
$532$	10.1716i	0.440994i
$533$	30.8284i	1.33533i
$534$	0	0
$535$	0	0
$536$	1.51472	0.0654259
$537$	0	0
$538$	38.1421i	1.64442i
$539$	6.82843	0.294121
$540$	0	0
$541$	11.3137	0.486414	0.243207	−	0.969974i	$-0.421801\pi$
$541$	11.3137	0.486414	0.243207	+	0.969974i	$0.421801\pi$
$542$	− 21.4853i	− 0.922872i
$543$	0	0
$544$	3.82843	0.164142
$545$	0	0
$546$	0	0
$547$	35.8701i	1.53369i	0.641831	+	0.766846i	$0.278175\pi$
$547$	35.8701i	1.53369i	−0.641831	+	0.766846i	$0.721825\pi$
$548$	− 46.4853i	− 1.98575i
$549$	0	0
$550$	0	0
$551$	−7.51472	−0.320138
$552$	0	0
$553$	− 5.48528i	− 0.233258i
$554$	11.6569	0.495252
$555$	0	0
$556$	−72.6274	−3.08009
$557$	5.17157i	0.219127i	0.993980	+	0.109563i	$0.0349452\pi$
$557$	5.17157i	0.219127i	−0.993980	+	0.109563i	$0.965055\pi$
$558$	0	0
$559$	−32.9706	−1.39451
$560$	0	0
$561$	0	0
$562$	77.4264i	3.26604i
$563$	15.3137i	0.645396i	0.946502	+	0.322698i	$0.104590\pi$
$563$	15.3137i	0.645396i	−0.946502	+	0.322698i	$0.895410\pi$
$564$	0	0
$565$	0	0
$566$	2.17157	0.0912780
$567$	0	0
$568$	− 34.5563i	− 1.44995i
$569$	15.2426	0.639005	0.319502	−	0.947585i	$-0.396484\pi$
$569$	15.2426	0.639005	0.319502	+	0.947585i	$0.396484\pi$
$570$	0	0
$571$	−9.02944	−0.377870	−0.188935	−	0.981990i	$-0.560504\pi$
$571$	−9.02944	−0.377870	−0.188935	+	0.981990i	$0.560504\pi$
$572$	− 10.8284i	− 0.452759i
$573$	0	0
$574$	10.8995	0.454936
$575$	0	0
$576$	0	0
$577$	23.9706i	0.997908i	0.866629	+	0.498954i	$0.166282\pi$
$577$	23.9706i	0.997908i	−0.866629	+	0.498954i	$0.833718\pi$
$578$	− 26.9706i	− 1.12183i
$579$	0	0
$580$	0	0
$581$	1.85786	0.0770772
$582$	0	0
$583$	7.65685i	0.317115i
$584$	−38.9706	−1.61261
$585$	0	0
$586$	42.4558	1.75383
$587$	36.6569i	1.51299i	0.653999	+	0.756495i	$0.273090\pi$
$587$	36.6569i	1.51299i	−0.653999	+	0.756495i	$0.726910\pi$
$588$	0	0
$589$	54.4264	2.24260
$590$	0	0
$591$	0	0
$592$	− 0.514719i	− 0.0211548i
$593$	3.79899i	0.156006i	0.996953	+	0.0780029i	$0.0248543\pi$
$593$	3.79899i	0.156006i	−0.996953	+	0.0780029i	$0.975146\pi$
$594$	0	0
$595$	0	0
$596$	−29.5858	−1.21188
$597$	0	0
$598$	6.82843i	0.279235i
$599$	36.3137	1.48374	0.741869	−	0.670545i	$-0.233940\pi$
$599$	36.3137	1.48374	0.741869	+	0.670545i	$0.233940\pi$
$600$	0	0
$601$	−14.8284	−0.604864	−0.302432	−	0.953171i	$-0.597799\pi$
$601$	−14.8284	−0.604864	−0.302432	+	0.953171i	$0.597799\pi$
$602$	11.6569i	0.475098i
$603$	0	0
$604$	−53.5980	−2.18087
$605$	0	0
$606$	0	0
$607$	− 2.97056i	− 0.120571i	−0.998181	−	0.0602857i	$-0.980799\pi$
$607$	− 2.97056i	− 0.120571i	0.998181	−	0.0602857i	$-0.0192012\pi$
$608$	− 10.1716i	− 0.412512i
$609$	0	0
$610$	0	0
$611$	−21.1716	−0.856510
$612$	0	0
$613$	42.0000i	1.69636i	0.529705	+	0.848182i	$0.322303\pi$
$613$	42.0000i	1.69636i	−0.529705	+	0.848182i	$0.677697\pi$
$614$	16.1421	0.651444
$615$	0	0
$616$	−1.82843	−0.0736694
$617$	9.85786i	0.396863i	0.980115	+	0.198431i	$0.0635847\pi$
$617$	9.85786i	0.396863i	−0.980115	+	0.198431i	$0.936415\pi$
$618$	0	0
$619$	−38.6274	−1.55257	−0.776283	−	0.630384i	$-0.782897\pi$
$619$	−38.6274	−1.55257	−0.776283	+	0.630384i	$0.782897\pi$
$620$	0	0
$621$	0	0
$622$	32.9706i	1.32200i
$623$	1.51472i	0.0606859i
$624$	0	0
$625$	0	0
$626$	65.5269	2.61898
$627$	0	0
$628$	− 22.9706i	− 0.916625i
$629$	−0.414214	−0.0165158
$630$	0	0
$631$	26.6274	1.06002	0.530010	−	0.847991i	$-0.322188\pi$
$631$	26.6274	1.06002	0.530010	+	0.847991i	$0.322188\pi$
$632$	− 58.4558i	− 2.32525i
$633$	0	0
$634$	−74.4264	−2.95585
$635$	0	0
$636$	0	0
$637$	19.3137i	0.765237i
$638$	− 2.82843i	− 0.111979i
$639$	0	0
$640$	0	0
$641$	42.4853	1.67807	0.839034	−	0.544079i	$-0.183121\pi$
$641$	42.4853	1.67807	0.839034	+	0.544079i	$0.183121\pi$
$642$	0	0
$643$	32.9706i	1.30023i	0.759835	+	0.650116i	$0.225280\pi$
$643$	32.9706i	1.30023i	−0.759835	+	0.650116i	$0.774720\pi$
$644$	1.58579	0.0624887
$645$	0	0
$646$	37.3848	1.47088
$647$	− 17.3431i	− 0.681829i	−0.940094	−	0.340915i	$-0.889263\pi$
$647$	− 17.3431i	− 0.681829i	0.940094	−	0.340915i	$-0.110737\pi$
$648$	0	0
$649$	11.0000	0.431788
$650$	0	0
$651$	0	0
$652$	60.4853i	2.36879i
$653$	− 40.4853i	− 1.58431i	−0.610319	−	0.792156i	$-0.708959\pi$
$653$	− 40.4853i	− 1.58431i	0.610319	−	0.792156i	$-0.291041\pi$
$654$	0	0
$655$	0	0
$656$	32.6985	1.27666
$657$	0	0
$658$	7.48528i	0.291807i
$659$	15.1127	0.588707	0.294354	−	0.955697i	$-0.404896\pi$
$659$	15.1127	0.588707	0.294354	+	0.955697i	$0.404896\pi$
$660$	0	0
$661$	−34.6569	−1.34800	−0.673998	−	0.738733i	$-0.735424\pi$
$661$	−34.6569	−1.34800	−0.673998	+	0.738733i	$0.735424\pi$
$662$	77.5980i	3.01593i
$663$	0	0
$664$	19.7990	0.768350
$665$	0	0
$666$	0	0
$667$	1.17157i	0.0453635i
$668$	83.4558i	3.22900i
$669$	0	0
$670$	0	0
$671$	8.82843	0.340818
$672$	0	0
$673$	− 11.6569i	− 0.449339i	−0.974435	−	0.224669i	$-0.927870\pi$
$673$	− 11.6569i	− 0.449339i	0.974435	−	0.224669i	$-0.0721302\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−19.1421	−0.736236
$677$	− 40.4853i	− 1.55598i	−0.628279	−	0.777988i	$-0.716240\pi$
$677$	− 40.4853i	− 1.55598i	0.628279	−	0.777988i	$-0.283760\pi$
$678$	0	0
$679$	−2.41421	−0.0926490
$680$	0	0
$681$	0	0
$682$	20.4853i	0.784422i
$683$	29.4853i	1.12822i	0.825699	+	0.564111i	$0.190781\pi$
$683$	29.4853i	1.12822i	−0.825699	+	0.564111i	$0.809219\pi$
$684$	0	0
$685$	0	0
$686$	13.8284	0.527972
$687$	0	0
$688$	34.9706i	1.33324i
$689$	−21.6569	−0.825060
$690$	0	0
$691$	15.4558	0.587968	0.293984	−	0.955810i	$-0.405019\pi$
$691$	15.4558	0.587968	0.293984	+	0.955810i	$0.405019\pi$
$692$	48.0711i	1.82739i
$693$	0	0
$694$	51.1127	1.94021
$695$	0	0
$696$	0	0
$697$	− 26.3137i	− 0.996703i
$698$	− 6.00000i	− 0.227103i
$699$	0	0
$700$	0	0
$701$	4.61522	0.174315	0.0871573	−	0.996195i	$-0.472222\pi$
$701$	4.61522	0.174315	0.0871573	+	0.996195i	$0.472222\pi$
$702$	0	0
$703$	1.10051i	0.0415063i
$704$	9.82843	0.370423
$705$	0	0
$706$	−10.8284	−0.407533
$707$	− 6.17157i	− 0.232106i
$708$	0	0
$709$	0.857864	0.0322178	0.0161089	−	0.999870i	$-0.494872\pi$
$709$	0.857864	0.0322178	0.0161089	+	0.999870i	$0.494872\pi$
$710$	0	0
$711$	0	0
$712$	16.1421i	0.604952i
$713$	− 8.48528i	− 0.317776i
$714$	0	0
$715$	0	0
$716$	−64.3137	−2.40352
$717$	0	0
$718$	− 37.4558i	− 1.39784i
$719$	−1.65685	−0.0617902	−0.0308951	−	0.999523i	$-0.509836\pi$
$719$	−1.65685	−0.0617902	−0.0308951	+	0.999523i	$0.509836\pi$
$720$	0	0
$721$	−5.65685	−0.210672
$722$	− 53.4558i	− 1.98942i
$723$	0	0
$724$	84.1127	3.12602
$725$	0	0
$726$	0	0
$727$	− 16.9706i	− 0.629403i	−0.949191	−	0.314702i	$-0.898096\pi$
$727$	− 16.9706i	− 0.629403i	0.949191	−	0.314702i	$-0.101904\pi$
$728$	− 5.17157i	− 0.191671i
$729$	0	0
$730$	0	0
$731$	28.1421	1.04087
$732$	0	0
$733$	− 3.85786i	− 0.142493i	−0.997459	−	0.0712467i	$-0.977302\pi$
$733$	− 3.85786i	− 0.142493i	0.997459	−	0.0712467i	$-0.0226978\pi$
$734$	3.17157	0.117065
$735$	0	0
$736$	−1.58579	−0.0584529
$737$	− 0.343146i	− 0.0126399i
$738$	0	0
$739$	−17.5858	−0.646904	−0.323452	−	0.946245i	$-0.604843\pi$
$739$	−17.5858	−0.646904	−0.323452	+	0.946245i	$0.604843\pi$
$740$	0	0
$741$	0	0
$742$	7.65685i	0.281092i
$743$	31.1127i	1.14141i	0.821154	+	0.570707i	$0.193331\pi$
$743$	31.1127i	1.14141i	−0.821154	+	0.570707i	$0.806669\pi$
$744$	0	0
$745$	0	0
$746$	57.1127	2.09104
$747$	0	0
$748$	9.24264i	0.337944i
$749$	−2.20101	−0.0804232
$750$	0	0
$751$	−47.5980	−1.73687	−0.868437	−	0.495799i	$-0.834875\pi$
$751$	−47.5980	−1.73687	−0.868437	+	0.495799i	$0.834875\pi$
$752$	22.4558i	0.818880i
$753$	0	0
$754$	8.00000	0.291343
$755$	0	0
$756$	0	0
$757$	− 33.3137i	− 1.21081i	−0.795919	−	0.605404i	$-0.793012\pi$
$757$	− 33.3137i	− 1.21081i	0.795919	−	0.605404i	$-0.206988\pi$
$758$	22.1421i	0.804239i
$759$	0	0
$760$	0	0
$761$	10.8284	0.392530	0.196265	−	0.980551i	$-0.437119\pi$
$761$	10.8284	0.392530	0.196265	+	0.980551i	$0.437119\pi$
$762$	0	0
$763$	− 2.20101i	− 0.0796819i
$764$	23.6274	0.854810
$765$	0	0
$766$	48.2843	1.74458
$767$	31.1127i	1.12341i
$768$	0	0
$769$	−3.65685	−0.131870	−0.0659348	−	0.997824i	$-0.521003\pi$
$769$	−3.65685	−0.131870	−0.0659348	+	0.997824i	$0.521003\pi$
$770$	0	0
$771$	0	0
$772$	12.6863i	0.456590i
$773$	4.82843i	0.173666i	0.996223	+	0.0868332i	$0.0276747\pi$
$773$	4.82843i	0.173666i	−0.996223	+	0.0868332i	$0.972325\pi$
$774$	0	0
$775$	0	0
$776$	−25.7279	−0.923579
$777$	0	0
$778$	42.6274i	1.52827i
$779$	−69.9117	−2.50485
$780$	0	0
$781$	−7.82843	−0.280123
$782$	− 5.82843i	− 0.208424i
$783$	0	0
$784$	20.4853	0.731617
$785$	0	0
$786$	0	0
$787$	− 22.0711i	− 0.786749i	−0.919378	−	0.393374i	$-0.871308\pi$
$787$	− 22.0711i	− 0.786749i	0.919378	−	0.393374i	$-0.128692\pi$
$788$	18.2132i	0.648819i
$789$	0	0
$790$	0	0
$791$	4.14214	0.147277
$792$	0	0
$793$	24.9706i	0.886731i
$794$	−86.7696	−3.07934
$795$	0	0
$796$	41.4558	1.46936
$797$	− 7.02944i	− 0.248995i	−0.992220	−	0.124498i	$-0.960268\pi$
$797$	− 7.02944i	− 0.248995i	0.992220	−	0.124498i	$-0.0397319\pi$
$798$	0	0
$799$	18.0711	0.639308
$800$	0	0
$801$	0	0
$802$	76.7696i	2.71083i
$803$	8.82843i	0.311548i
$804$	0	0
$805$	0	0
$806$	−57.9411	−2.04089
$807$	0	0
$808$	− 65.7696i	− 2.31376i
$809$	−17.7279	−0.623281	−0.311640	−	0.950200i	$-0.600878\pi$
$809$	−17.7279	−0.623281	−0.311640	+	0.950200i	$0.600878\pi$
$810$	0	0
$811$	−32.2132	−1.13116	−0.565579	−	0.824694i	$-0.691347\pi$
$811$	−32.2132	−1.13116	−0.565579	+	0.824694i	$0.691347\pi$
$812$	− 1.85786i	− 0.0651983i
$813$	0	0
$814$	−0.414214	−0.0145182
$815$	0	0
$816$	0	0
$817$	− 74.7696i	− 2.61586i
$818$	− 10.0000i	− 0.349642i
$819$	0	0
$820$	0	0
$821$	23.7990	0.830590	0.415295	−	0.909687i	$-0.363678\pi$
$821$	23.7990	0.830590	0.415295	+	0.909687i	$0.363678\pi$
$822$	0	0
$823$	− 18.9706i	− 0.661272i	−0.943758	−	0.330636i	$-0.892737\pi$
$823$	− 18.9706i	− 0.661272i	0.943758	−	0.330636i	$-0.107263\pi$
$824$	−60.2843	−2.10010
$825$	0	0
$826$	11.0000	0.382739
$827$	35.3137i	1.22798i	0.789315	+	0.613989i	$0.210436\pi$
$827$	35.3137i	1.22798i	−0.789315	+	0.613989i	$0.789564\pi$
$828$	0	0
$829$	−19.9411	−0.692584	−0.346292	−	0.938127i	$-0.612559\pi$
$829$	−19.9411	−0.692584	−0.346292	+	0.938127i	$0.612559\pi$
$830$	0	0
$831$	0	0
$832$	27.7990i	0.963757i
$833$	− 16.4853i	− 0.571181i
$834$	0	0
$835$	0	0
$836$	24.5563	0.849299
$837$	0	0
$838$	61.5269i	2.12541i
$839$	−24.6863	−0.852265	−0.426133	−	0.904661i	$-0.640124\pi$
$839$	−24.6863	−0.852265	−0.426133	+	0.904661i	$0.640124\pi$
$840$	0	0
$841$	−27.6274	−0.952670
$842$	65.1838i	2.24638i
$843$	0	0
$844$	50.9706	1.75448
$845$	0	0
$846$	0	0
$847$	0.414214i	0.0142325i
$848$	22.9706i	0.788812i
$849$	0	0
$850$	0	0
$851$	0.171573	0.00588144
$852$	0	0
$853$	24.8284i	0.850109i	0.905168	+	0.425055i	$0.139745\pi$
$853$	24.8284i	0.850109i	−0.905168	+	0.425055i	$0.860255\pi$
$854$	8.82843	0.302103
$855$	0	0
$856$	−23.4558	−0.801704
$857$	− 22.6985i	− 0.775365i	−0.921793	−	0.387683i	$-0.873276\pi$
$857$	− 22.6985i	− 0.775365i	0.921793	−	0.387683i	$-0.126724\pi$
$858$	0	0
$859$	3.51472	0.119921	0.0599603	−	0.998201i	$-0.480903\pi$
$859$	3.51472	0.119921	0.0599603	+	0.998201i	$0.480903\pi$
$860$	0	0
$861$	0	0
$862$	2.82843i	0.0963366i
$863$	− 51.3137i	− 1.74674i	−0.487058	−	0.873369i	$-0.661930\pi$
$863$	− 51.3137i	− 1.74674i	0.487058	−	0.873369i	$-0.338070\pi$
$864$	0	0
$865$	0	0
$866$	32.1421	1.09223
$867$	0	0
$868$	13.4558i	0.456721i
$869$	−13.2426	−0.449226
$870$	0	0
$871$	0.970563	0.0328863
$872$	− 23.4558i	− 0.794315i
$873$	0	0
$874$	−15.4853	−0.523797
$875$	0	0
$876$	0	0
$877$	− 47.1127i	− 1.59088i	−0.606031	−	0.795441i	$-0.707239\pi$
$877$	− 47.1127i	− 1.59088i	0.606031	−	0.795441i	$-0.292761\pi$
$878$	5.48528i	0.185119i
$879$	0	0
$880$	0	0
$881$	24.0000	0.808581	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808581	0.404290	+	0.914631i	$0.367519\pi$
$882$	0	0
$883$	6.62742i	0.223030i	0.993763	+	0.111515i	$0.0355704\pi$
$883$	6.62742i	0.223030i	−0.993763	+	0.111515i	$0.964430\pi$
$884$	−26.1421	−0.879255
$885$	0	0
$886$	−19.2426	−0.646469
$887$	− 6.14214i	− 0.206233i	−0.994669	−	0.103116i	$-0.967119\pi$
$887$	− 6.14214i	− 0.206233i	0.994669	−	0.103116i	$-0.0328814\pi$
$888$	0	0
$889$	−3.00000	−0.100617
$890$	0	0
$891$	0	0
$892$	− 81.0538i	− 2.71388i
$893$	− 48.0122i	− 1.60667i
$894$	0	0
$895$	0	0
$896$	8.51472	0.284457
$897$	0	0
$898$	15.6569i	0.522476i
$899$	−9.94113	−0.331555
$900$	0	0
$901$	18.4853	0.615834
$902$	− 26.3137i	− 0.876151i
$903$	0	0
$904$	44.1421	1.46815
$905$	0	0
$906$	0	0
$907$	14.4853i	0.480976i	0.970652	+	0.240488i	$0.0773074\pi$
$907$	14.4853i	0.480976i	−0.970652	+	0.240488i	$0.922693\pi$
$908$	− 70.7696i	− 2.34857i
$909$	0	0
$910$	0	0
$911$	−45.4853	−1.50699	−0.753497	−	0.657451i	$-0.771635\pi$
$911$	−45.4853	−1.50699	−0.753497	+	0.657451i	$0.771635\pi$
$912$	0	0
$913$	− 4.48528i	− 0.148441i
$914$	9.31371	0.308070
$915$	0	0
$916$	−9.62742	−0.318099
$917$	− 2.82843i	− 0.0934029i
$918$	0	0
$919$	−44.2132	−1.45846	−0.729230	−	0.684269i	$-0.760121\pi$
$919$	−44.2132	−1.45846	−0.729230	+	0.684269i	$0.760121\pi$
$920$	0	0
$921$	0	0
$922$	− 79.1127i	− 2.60544i
$923$	− 22.1421i	− 0.728817i
$924$	0	0
$925$	0	0
$926$	−84.4264	−2.77442
$927$	0	0
$928$	1.85786i	0.0609874i
$929$	25.7990	0.846437	0.423219	−	0.906028i	$-0.360900\pi$
$929$	25.7990	0.846437	0.423219	+	0.906028i	$0.360900\pi$
$930$	0	0
$931$	−43.7990	−1.43545
$932$	− 63.3848i	− 2.07624i
$933$	0	0
$934$	−25.6569	−0.839518
$935$	0	0
$936$	0	0
$937$	− 16.0000i	− 0.522697i	−0.965244	−	0.261349i	$-0.915833\pi$
$937$	− 16.0000i	− 0.522697i	0.965244	−	0.261349i	$-0.0841672\pi$
$938$	− 0.343146i	− 0.0112041i
$939$	0	0
$940$	0	0
$941$	3.10051	0.101074	0.0505368	−	0.998722i	$-0.483907\pi$
$941$	3.10051	0.101074	0.0505368	+	0.998722i	$0.483907\pi$
$942$	0	0
$943$	10.8995i	0.354936i
$944$	33.0000	1.07406
$945$	0	0
$946$	28.1421	0.914980
$947$	− 2.79899i	− 0.0909549i	−0.998965	−	0.0454775i	$-0.985519\pi$
$947$	− 2.79899i	− 0.0909549i	0.998965	−	0.0454775i	$-0.0144809\pi$
$948$	0	0
$949$	−24.9706	−0.810579
$950$	0	0
$951$	0	0
$952$	4.41421i	0.143065i
$953$	19.0416i	0.616819i	0.951254	+	0.308409i	$0.0997967\pi$
$953$	19.0416i	0.616819i	−0.951254	+	0.308409i	$0.900203\pi$
$954$	0	0
$955$	0	0
$956$	−90.5685	−2.92920
$957$	0	0
$958$	59.1127i	1.90984i
$959$	−5.02944	−0.162409
$960$	0	0
$961$	41.0000	1.32258
$962$	− 1.17157i	− 0.0377730i
$963$	0	0
$964$	−54.1421	−1.74380
$965$	0	0
$966$	0	0
$967$	38.0000i	1.22200i	0.791632	+	0.610999i	$0.209232\pi$
$967$	38.0000i	1.22200i	−0.791632	+	0.610999i	$0.790768\pi$
$968$	4.41421i	0.141878i
$969$	0	0
$970$	0	0
$971$	27.6863	0.888495	0.444248	−	0.895904i	$-0.353471\pi$
$971$	27.6863	0.888495	0.444248	+	0.895904i	$0.353471\pi$
$972$	0	0
$973$	7.85786i	0.251912i
$974$	−15.6569	−0.501678
$975$	0	0
$976$	26.4853	0.847773
$977$	52.5685i	1.68182i	0.541178	+	0.840908i	$0.317979\pi$
$977$	52.5685i	1.68182i	−0.541178	+	0.840908i	$0.682021\pi$
$978$	0	0
$979$	3.65685	0.116874
$980$	0	0
$981$	0	0
$982$	− 58.2843i	− 1.85993i
$983$	− 5.28427i	− 0.168542i	−0.996443	−	0.0842710i	$-0.973144\pi$
$983$	− 5.28427i	− 0.168542i	0.996443	−	0.0842710i	$-0.0268562\pi$
$984$	0	0
$985$	0	0
$986$	−6.82843	−0.217461
$987$	0	0
$988$	69.4558i	2.20968i
$989$	−11.6569	−0.370666
$990$	0	0
$991$	14.2843	0.453755	0.226877	−	0.973923i	$-0.427148\pi$
$991$	14.2843	0.453755	0.226877	+	0.973923i	$0.427148\pi$
$992$	− 13.4558i	− 0.427223i
$993$	0	0
$994$	−7.82843	−0.248303
$995$	0	0
$996$	0	0
$997$	43.2548i	1.36989i	0.728593	+	0.684947i	$0.240175\pi$
$997$	43.2548i	1.36989i	−0.728593	+	0.684947i	$0.759825\pi$
$998$	84.9117i	2.68783i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.c.o.199.1		4
3.2	odd	2		825.2.c.d.199.4		4
5.2	odd	4		2475.2.a.w.1.2		2
5.3	odd	4		2475.2.a.l.1.1		2
5.4	even	2	inner	2475.2.c.o.199.4		4
15.2	even	4		825.2.a.d.1.1	✓	2
15.8	even	4		825.2.a.f.1.2	yes	2
15.14	odd	2		825.2.c.d.199.1		4
165.32	odd	4		9075.2.a.ca.1.2		2
165.98	odd	4		9075.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.a.d.1.1	✓	2	15.2	even	4
825.2.a.f.1.2	yes	2	15.8	even	4
825.2.c.d.199.1		4	15.14	odd	2
825.2.c.d.199.4		4	3.2	odd	2
2475.2.a.l.1.1		2	5.3	odd	4
2475.2.a.w.1.2		2	5.2	odd	4
2475.2.c.o.199.1		4	1.1	even	1	trivial
2475.2.c.o.199.4		4	5.4	even	2	inner
9075.2.a.w.1.1		2	165.98	odd	4
9075.2.a.ca.1.2		2	165.32	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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.41421i q^{2} -3.82843 q^{4} +0.414214i q^{7} +4.41421i q^{8} +O(q^{10})\)	\(q-2.41421i q^{2} -3.82843 q^{4} +0.414214i q^{7} +4.41421i q^{8} +1.00000 q^{11} +2.82843i q^{13} +1.00000 q^{14} +3.00000 q^{16} -2.41421i q^{17} -6.41421 q^{19} -2.41421i q^{22} +1.00000i q^{23} +6.82843 q^{26} -1.58579i q^{28} +1.17157 q^{29} -8.48528 q^{31} +1.58579i q^{32} -5.82843 q^{34} -0.171573i q^{37} +15.4853i q^{38} +10.8995 q^{41} +11.6569i q^{43} -3.82843 q^{44} +2.41421 q^{46} +7.48528i q^{47} +6.82843 q^{49} -10.8284i q^{52} +7.65685i q^{53} -1.82843 q^{56} -2.82843i q^{58} +11.0000 q^{59} +8.82843 q^{61} +20.4853i q^{62} +9.82843 q^{64} -0.343146i q^{67} +9.24264i q^{68} -7.82843 q^{71} +8.82843i q^{73} -0.414214 q^{74} +24.5563 q^{76} +0.414214i q^{77} -13.2426 q^{79} -26.3137i q^{82} -4.48528i q^{83} +28.1421 q^{86} +4.41421i q^{88} +3.65685 q^{89} -1.17157 q^{91} -3.82843i q^{92} +18.0711 q^{94} +5.82843i q^{97} -16.4853i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{11} + 4 q^{14} + 12 q^{16} - 20 q^{19} + 16 q^{26} + 16 q^{29} - 12 q^{34} + 4 q^{41} - 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{56} + 44 q^{59} + 24 q^{61} + 28 q^{64} - 20 q^{71} + 4 q^{74} + 36 q^{76} - 36 q^{79} + 56 q^{86} - 8 q^{89} - 16 q^{91} + 44 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^11 + 4 * q^14 + 12 * q^16 - 20 * q^19 + 16 * q^26 + 16 * q^29 - 12 * q^34 + 4 * q^41 - 4 * q^44 + 4 * q^46 + 16 * q^49 + 4 * q^56 + 44 * q^59 + 24 * q^61 + 28 * q^64 - 20 * q^71 + 4 * q^74 + 36 * q^76 - 36 * q^79 + 56 * q^86 - 8 * q^89 - 16 * q^91 + 44 * q^94

Introduction

L-functions

Modular forms

Varieties

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Database

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