LMFDB - Embedded newform 5625.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

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:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.46840000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 $ x^6 - x^5 - 11x^4 + 8x^3 + 31x^2 - 15x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.78712$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.78712 q^{2} +5.76803 q^{4} -3.15000 q^{7} -10.5020 q^{8} +O(q^{10})$	$q-2.78712 q^{2} +5.76803 q^{4} -3.15000 q^{7} -10.5020 q^{8} -2.94681 q^{11} +0.188171 q^{13} +8.77943 q^{14} +17.7341 q^{16} -2.62743 q^{17} +1.94100 q^{19} +8.21310 q^{22} +1.30447 q^{23} -0.524454 q^{26} -18.1693 q^{28} -1.23197 q^{29} +2.65174 q^{31} -28.4233 q^{32} +7.32297 q^{34} +10.1161 q^{37} -5.40980 q^{38} -3.85940 q^{41} -9.63734 q^{43} -16.9973 q^{44} -3.63570 q^{46} +11.9846 q^{47} +2.92250 q^{49} +1.08537 q^{52} -0.369521 q^{53} +33.0812 q^{56} +3.43364 q^{58} +6.90864 q^{59} +5.68643 q^{61} -7.39071 q^{62} +43.7507 q^{64} +3.24188 q^{67} -15.1551 q^{68} -6.69982 q^{71} -9.46938 q^{73} -28.1948 q^{74} +11.1957 q^{76} +9.28244 q^{77} +0.420137 q^{79} +10.7566 q^{82} +12.1635 q^{83} +26.8604 q^{86} +30.9473 q^{88} +15.1828 q^{89} -0.592737 q^{91} +7.52421 q^{92} -33.4025 q^{94} +13.9035 q^{97} -8.14536 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8	$ 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38} - 33 q^{41} - 3 q^{43} + 11 q^{44} - 12 q^{46} + 6 q^{47} + 4 q^{49} - 33 q^{52} - 14 q^{53} + 30 q^{56} - q^{58} + 8 q^{59} + 34 q^{61} - 31 q^{62} + 12 q^{64} + 2 q^{67} - 27 q^{68} + 3 q^{71} - 36 q^{73} - 36 q^{74} + 27 q^{76} - 16 q^{77} + 25 q^{79} + 36 q^{82} + 12 q^{83} + 30 q^{86} + 56 q^{88} - 18 q^{89} + 28 q^{91} - 3 q^{92} - 50 q^{94} + 7 q^{97} - 15 q^{98}+O(q^{100}) $ 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8 + 4 * q^14 + 17 * q^16 - 2 * q^17 - 2 * q^19 + 9 * q^22 + q^23 - 37 * q^26 - 44 * q^28 - 31 * q^29 - 2 * q^31 - 33 * q^32 + 37 * q^34 - 22 * q^37 - 27 * q^38 - 33 * q^41 - 3 * q^43 + 11 * q^44 - 12 * q^46 + 6 * q^47 + 4 * q^49 - 33 * q^52 - 14 * q^53 + 30 * q^56 - q^58 + 8 * q^59 + 34 * q^61 - 31 * q^62 + 12 * q^64 + 2 * q^67 - 27 * q^68 + 3 * q^71 - 36 * q^73 - 36 * q^74 + 27 * q^76 - 16 * q^77 + 25 * q^79 + 36 * q^82 + 12 * q^83 + 30 * q^86 + 56 * q^88 - 18 * q^89 + 28 * q^91 - 3 * q^92 - 50 * q^94 + 7 * q^97 - 15 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−2.78712	−1.97079	−0.985395	−	0.170281i	$-0.945532\pi$
$2$	−2.78712	−1.97079	−0.985395	+	0.170281i	$0.945532\pi$
$3$	0	0
$3$	0	0
$4$	5.76803	2.88402
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.15000	−1.19059	−0.595294	−	0.803508i	$-0.702964\pi$
$7$	−3.15000	−1.19059	−0.595294	+	0.803508i	$0.702964\pi$
$8$	−10.5020	−3.71300
$9$	0	0
$10$	0	0
$11$	−2.94681	−0.888496	−0.444248	−	0.895904i	$-0.646529\pi$
$11$	−2.94681	−0.888496	−0.444248	+	0.895904i	$0.646529\pi$
$12$	0	0
$13$	0.188171	0.0521891	0.0260946	−	0.999659i	$-0.491693\pi$
$13$	0.188171	0.0521891	0.0260946	+	0.999659i	$0.491693\pi$
$14$	8.77943	2.34640
$15$	0	0
$16$	17.7341	4.43354
$17$	−2.62743	−0.637246	−0.318623	−	0.947882i	$-0.603220\pi$
$17$	−2.62743	−0.637246	−0.318623	+	0.947882i	$0.603220\pi$
$18$	0	0
$19$	1.94100	0.445296	0.222648	−	0.974899i	$-0.428530\pi$
$19$	1.94100	0.445296	0.222648	+	0.974899i	$0.428530\pi$
$20$	0	0
$21$	0	0
$22$	8.21310	1.75104
$23$	1.30447	0.272000	0.136000	−	0.990709i	$-0.456575\pi$
$23$	1.30447	0.272000	0.136000	+	0.990709i	$0.456575\pi$
$24$	0	0
$25$	0	0
$26$	−0.524454	−0.102854
$27$	0	0
$28$	−18.1693	−3.43368
$29$	−1.23197	−0.228770	−0.114385	−	0.993436i	$-0.536490\pi$
$29$	−1.23197	−0.228770	−0.114385	+	0.993436i	$0.536490\pi$
$30$	0	0
$31$	2.65174	0.476266	0.238133	−	0.971233i	$-0.423465\pi$
$31$	2.65174	0.476266	0.238133	+	0.971233i	$0.423465\pi$
$32$	−28.4233	−5.02457
$33$	0	0
$34$	7.32297	1.25588
$35$	0	0
$36$	0	0
$37$	10.1161	1.66308	0.831540	−	0.555466i	$-0.187460\pi$
$37$	10.1161	1.66308	0.831540	+	0.555466i	$0.187460\pi$
$38$	−5.40980	−0.877585
$39$	0	0
$40$	0	0
$41$	−3.85940	−0.602737	−0.301368	−	0.953508i	$-0.597443\pi$
$41$	−3.85940	−0.602737	−0.301368	+	0.953508i	$0.597443\pi$
$42$	0	0
$43$	−9.63734	−1.46968	−0.734840	−	0.678240i	$-0.762743\pi$
$43$	−9.63734	−1.46968	−0.734840	+	0.678240i	$0.762743\pi$
$44$	−16.9973	−2.56244
$45$	0	0
$46$	−3.63570	−0.536055
$47$	11.9846	1.74814	0.874068	−	0.485804i	$-0.161473\pi$
$47$	11.9846	1.74814	0.874068	+	0.485804i	$0.161473\pi$
$48$	0	0
$49$	2.92250	0.417500
$50$	0	0
$51$	0	0
$52$	1.08537	0.150514
$53$	−0.369521	−0.0507576	−0.0253788	−	0.999678i	$-0.508079\pi$
$53$	−0.369521	−0.0507576	−0.0253788	+	0.999678i	$0.508079\pi$
$54$	0	0
$55$	0	0
$56$	33.0812	4.42066
$57$	0	0
$58$	3.43364	0.450859
$59$	6.90864	0.899428	0.449714	−	0.893173i	$-0.351526\pi$
$59$	6.90864	0.899428	0.449714	+	0.893173i	$0.351526\pi$
$60$	0	0
$61$	5.68643	0.728073	0.364037	−	0.931385i	$-0.381398\pi$
$61$	5.68643	0.728073	0.364037	+	0.931385i	$0.381398\pi$
$62$	−7.39071	−0.938621
$63$	0	0
$64$	43.7507	5.46884
$65$	0	0
$66$	0	0
$67$	3.24188	0.396058	0.198029	−	0.980196i	$-0.436546\pi$
$67$	3.24188	0.396058	0.198029	+	0.980196i	$0.436546\pi$
$68$	−15.1551	−1.83783
$69$	0	0
$70$	0	0
$71$	−6.69982	−0.795122	−0.397561	−	0.917576i	$-0.630143\pi$
$71$	−6.69982	−0.795122	−0.397561	+	0.917576i	$0.630143\pi$
$72$	0	0
$73$	−9.46938	−1.10831	−0.554153	−	0.832415i	$-0.686958\pi$
$73$	−9.46938	−1.10831	−0.554153	+	0.832415i	$0.686958\pi$
$74$	−28.1948	−3.27758
$75$	0	0
$76$	11.1957	1.28424
$77$	9.28244	1.05783
$78$	0	0
$79$	0.420137	0.0472691	0.0236345	−	0.999721i	$-0.492476\pi$
$79$	0.420137	0.0472691	0.0236345	+	0.999721i	$0.492476\pi$
$80$	0	0
$81$	0	0
$82$	10.7566	1.18787
$83$	12.1635	1.33512	0.667559	−	0.744557i	$-0.267339\pi$
$83$	12.1635	1.33512	0.667559	+	0.744557i	$0.267339\pi$
$84$	0	0
$85$	0	0
$86$	26.8604	2.89643
$87$	0	0
$88$	30.9473	3.29899
$89$	15.1828	1.60937	0.804687	−	0.593699i	$-0.202333\pi$
$89$	15.1828	1.60937	0.804687	+	0.593699i	$0.202333\pi$
$90$	0	0
$91$	−0.592737	−0.0621358
$92$	7.52421	0.784453
$93$	0	0
$94$	−33.4025	−3.44521
$95$	0	0
$96$	0	0
$97$	13.9035	1.41169	0.705845	−	0.708367i	$-0.250568\pi$
$97$	13.9035	1.41169	0.705845	+	0.708367i	$0.250568\pi$
$98$	−8.14536	−0.822805
$99$	0	0
$100$	0	0
$101$	9.67188	0.962388	0.481194	−	0.876614i	$-0.340203\pi$
$101$	9.67188	0.962388	0.481194	+	0.876614i	$0.340203\pi$
$102$	0	0
$103$	8.69434	0.856679	0.428339	−	0.903618i	$-0.359099\pi$
$103$	8.69434	0.856679	0.428339	+	0.903618i	$0.359099\pi$
$104$	−1.97616	−0.193778
$105$	0	0
$106$	1.02990	0.100033
$107$	2.56585	0.248050	0.124025	−	0.992279i	$-0.460420\pi$
$107$	2.56585	0.248050	0.124025	+	0.992279i	$0.460420\pi$
$108$	0	0
$109$	−15.2352	−1.45926	−0.729632	−	0.683840i	$-0.760309\pi$
$109$	−15.2352	−1.45926	−0.729632	+	0.683840i	$0.760309\pi$
$110$	0	0
$111$	0	0
$112$	−55.8626	−5.27852
$113$	−2.20391	−0.207326	−0.103663	−	0.994612i	$-0.533056\pi$
$113$	−2.20391	−0.207326	−0.103663	+	0.994612i	$0.533056\pi$
$114$	0	0
$115$	0	0
$116$	−7.10602	−0.659778
$117$	0	0
$118$	−19.2552	−1.77258
$119$	8.27641	0.758697
$120$	0	0
$121$	−2.31633	−0.210575
$122$	−15.8488	−1.43488
$123$	0	0
$124$	15.2953	1.37356
$125$	0	0
$126$	0	0
$127$	−14.9689	−1.32827	−0.664136	−	0.747611i	$-0.731201\pi$
$127$	−14.9689	−1.32827	−0.664136	+	0.747611i	$0.731201\pi$
$128$	−65.0920	−5.75337
$129$	0	0
$130$	0	0
$131$	−11.2957	−0.986911	−0.493456	−	0.869771i	$-0.664266\pi$
$131$	−11.2957	−0.986911	−0.493456	+	0.869771i	$0.664266\pi$
$132$	0	0
$133$	−6.11415	−0.530164
$134$	−9.03550	−0.780548
$135$	0	0
$136$	27.5932	2.36610
$137$	−4.17877	−0.357017	−0.178508	−	0.983938i	$-0.557127\pi$
$137$	−4.17877	−0.357017	−0.178508	+	0.983938i	$0.557127\pi$
$138$	0	0
$139$	−15.3552	−1.30241	−0.651207	−	0.758900i	$-0.725737\pi$
$139$	−15.3552	−1.30241	−0.651207	+	0.758900i	$0.725737\pi$
$140$	0	0
$141$	0	0
$142$	18.6732	1.56702
$143$	−0.554502	−0.0463698
$144$	0	0
$145$	0	0
$146$	26.3923	2.18424
$147$	0	0
$148$	58.3501	4.79635
$149$	−3.43364	−0.281294	−0.140647	−	0.990060i	$-0.544918\pi$
$149$	−3.43364	−0.281294	−0.140647	+	0.990060i	$0.544918\pi$
$150$	0	0
$151$	10.7716	0.876581	0.438290	−	0.898833i	$-0.355584\pi$
$151$	10.7716	0.876581	0.438290	+	0.898833i	$0.355584\pi$
$152$	−20.3843	−1.65338
$153$	0	0
$154$	−25.8713	−2.08477
$155$	0	0
$156$	0	0
$157$	−13.4045	−1.06980	−0.534900	−	0.844916i	$-0.679651\pi$
$157$	−13.4045	−1.06980	−0.534900	+	0.844916i	$0.679651\pi$
$158$	−1.17097	−0.0931574
$159$	0	0
$160$	0	0
$161$	−4.10907	−0.323840
$162$	0	0
$163$	−15.4178	−1.20762	−0.603808	−	0.797129i	$-0.706351\pi$
$163$	−15.4178	−1.20762	−0.603808	+	0.797129i	$0.706351\pi$
$164$	−22.2611	−1.73830
$165$	0	0
$166$	−33.9011	−2.63124
$167$	−17.4045	−1.34680	−0.673402	−	0.739276i	$-0.735168\pi$
$167$	−17.4045	−1.34680	−0.673402	+	0.739276i	$0.735168\pi$
$168$	0	0
$169$	−12.9646	−0.997276
$170$	0	0
$171$	0	0
$172$	−55.5885	−4.23858
$173$	19.2737	1.46535	0.732676	−	0.680578i	$-0.238271\pi$
$173$	19.2737	1.46535	0.732676	+	0.680578i	$0.238271\pi$
$174$	0	0
$175$	0	0
$176$	−52.2591	−3.93918
$177$	0	0
$178$	−42.3163	−3.17174
$179$	−23.7940	−1.77845	−0.889223	−	0.457475i	$-0.848754\pi$
$179$	−23.7940	−1.77845	−0.889223	+	0.457475i	$0.848754\pi$
$180$	0	0
$181$	−10.6891	−0.794516	−0.397258	−	0.917707i	$-0.630038\pi$
$181$	−10.6891	−0.794516	−0.397258	+	0.917707i	$0.630038\pi$
$182$	1.65203	0.122457
$183$	0	0
$184$	−13.6995	−1.00994
$185$	0	0
$186$	0	0
$187$	7.74253	0.566190
$188$	69.1277	5.04165
$189$	0	0
$190$	0	0
$191$	14.3581	1.03892	0.519458	−	0.854496i	$-0.326134\pi$
$191$	14.3581	1.03892	0.519458	+	0.854496i	$0.326134\pi$
$192$	0	0
$193$	1.95508	0.140730	0.0703648	−	0.997521i	$-0.477584\pi$
$193$	1.95508	0.140730	0.0703648	+	0.997521i	$0.477584\pi$
$194$	−38.7508	−2.78214
$195$	0	0
$196$	16.8571	1.20408
$197$	1.17958	0.0840417	0.0420209	−	0.999117i	$-0.486620\pi$
$197$	1.17958	0.0840417	0.0420209	+	0.999117i	$0.486620\pi$
$198$	0	0
$199$	−6.55820	−0.464899	−0.232449	−	0.972608i	$-0.574674\pi$
$199$	−6.55820	−0.464899	−0.232449	+	0.972608i	$0.574674\pi$
$200$	0	0
$201$	0	0
$202$	−26.9567	−1.89666
$203$	3.88069	0.272371
$204$	0	0
$205$	0	0
$206$	−24.2322	−1.68833
$207$	0	0
$208$	3.33705	0.231382
$209$	−5.71975	−0.395643
$210$	0	0
$211$	12.3573	0.850715	0.425357	−	0.905026i	$-0.360148\pi$
$211$	12.3573	0.850715	0.425357	+	0.905026i	$0.360148\pi$
$212$	−2.13141	−0.146386
$213$	0	0
$214$	−7.15134	−0.488856
$215$	0	0
$216$	0	0
$217$	−8.35298	−0.567037
$218$	42.4622	2.87591
$219$	0	0
$220$	0	0
$221$	−0.494405	−0.0332573
$222$	0	0
$223$	−7.30693	−0.489308	−0.244654	−	0.969610i	$-0.578674\pi$
$223$	−7.30693	−0.489308	−0.244654	+	0.969610i	$0.578674\pi$
$224$	89.5333	5.98219
$225$	0	0
$226$	6.14256	0.408597
$227$	21.0202	1.39516	0.697579	−	0.716508i	$-0.254261\pi$
$227$	21.0202	1.39516	0.697579	+	0.716508i	$0.254261\pi$
$228$	0	0
$229$	19.5544	1.29219	0.646094	−	0.763258i	$-0.276401\pi$
$229$	19.5544	1.29219	0.646094	+	0.763258i	$0.276401\pi$
$230$	0	0
$231$	0	0
$232$	12.9381	0.849425
$233$	−12.1472	−0.795792	−0.397896	−	0.917430i	$-0.630260\pi$
$233$	−12.1472	−0.795792	−0.397896	+	0.917430i	$0.630260\pi$
$234$	0	0
$235$	0	0
$236$	39.8493	2.59397
$237$	0	0
$238$	−23.0673	−1.49523
$239$	−13.4517	−0.870119	−0.435059	−	0.900402i	$-0.643273\pi$
$239$	−13.4517	−0.870119	−0.435059	+	0.900402i	$0.643273\pi$
$240$	0	0
$241$	19.9370	1.28426	0.642129	−	0.766597i	$-0.278051\pi$
$241$	19.9370	1.28426	0.642129	+	0.766597i	$0.278051\pi$
$242$	6.45588	0.415000
$243$	0	0
$244$	32.7995	2.09978
$245$	0	0
$246$	0	0
$247$	0.365239	0.0232396
$248$	−27.8485	−1.76838
$249$	0	0
$250$	0	0
$251$	2.76013	0.174218	0.0871088	−	0.996199i	$-0.472237\pi$
$251$	2.76013	0.174218	0.0871088	+	0.996199i	$0.472237\pi$
$252$	0	0
$253$	−3.84401	−0.241671
$254$	41.7200	2.61775
$255$	0	0
$256$	93.9177	5.86986
$257$	17.2848	1.07820	0.539098	−	0.842243i	$-0.318765\pi$
$257$	17.2848	1.07820	0.539098	+	0.842243i	$0.318765\pi$
$258$	0	0
$259$	−31.8658	−1.98004
$260$	0	0
$261$	0	0
$262$	31.4825	1.94500
$263$	3.96578	0.244541	0.122270	−	0.992497i	$-0.460983\pi$
$263$	3.96578	0.244541	0.122270	+	0.992497i	$0.460983\pi$
$264$	0	0
$265$	0	0
$266$	17.0409	1.04484
$267$	0	0
$268$	18.6993	1.14224
$269$	−1.19564	−0.0728993	−0.0364496	−	0.999335i	$-0.511605\pi$
$269$	−1.19564	−0.0728993	−0.0364496	+	0.999335i	$0.511605\pi$
$270$	0	0
$271$	−2.18295	−0.132605	−0.0663023	−	0.997800i	$-0.521120\pi$
$271$	−2.18295	−0.132605	−0.0663023	+	0.997800i	$0.521120\pi$
$272$	−46.5953	−2.82525
$273$	0	0
$274$	11.6467	0.703605
$275$	0	0
$276$	0	0
$277$	1.95174	0.117268	0.0586342	−	0.998280i	$-0.481325\pi$
$277$	1.95174	0.117268	0.0586342	+	0.998280i	$0.481325\pi$
$278$	42.7969	2.56679
$279$	0	0
$280$	0	0
$281$	−2.30131	−0.137284	−0.0686422	−	0.997641i	$-0.521867\pi$
$281$	−2.30131	−0.137284	−0.0686422	+	0.997641i	$0.521867\pi$
$282$	0	0
$283$	−10.6459	−0.632835	−0.316417	−	0.948620i	$-0.602480\pi$
$283$	−10.6459	−0.632835	−0.316417	+	0.948620i	$0.602480\pi$
$284$	−38.6448	−2.29315
$285$	0	0
$286$	1.54546	0.0913852
$287$	12.1571	0.717611
$288$	0	0
$289$	−10.0966	−0.593918
$290$	0	0
$291$	0	0
$292$	−54.6197	−3.19638
$293$	−14.9591	−0.873921	−0.436960	−	0.899481i	$-0.643945\pi$
$293$	−14.9591	−0.873921	−0.436960	+	0.899481i	$0.643945\pi$
$294$	0	0
$295$	0	0
$296$	−106.239	−6.17502
$297$	0	0
$298$	9.56995	0.554373
$299$	0.245462	0.0141954
$300$	0	0
$301$	30.3576	1.74978
$302$	−30.0217	−1.72756
$303$	0	0
$304$	34.4220	1.97424
$305$	0	0
$306$	0	0
$307$	−3.07333	−0.175404	−0.0877021	−	0.996147i	$-0.527952\pi$
$307$	−3.07333	−0.175404	−0.0877021	+	0.996147i	$0.527952\pi$
$308$	53.5414	3.05081
$309$	0	0
$310$	0	0
$311$	−24.9095	−1.41249	−0.706244	−	0.707968i	$-0.749612\pi$
$311$	−24.9095	−1.41249	−0.706244	+	0.707968i	$0.749612\pi$
$312$	0	0
$313$	9.28562	0.524854	0.262427	−	0.964952i	$-0.415477\pi$
$313$	9.28562	0.524854	0.262427	+	0.964952i	$0.415477\pi$
$314$	37.3601	2.10835
$315$	0	0
$316$	2.42336	0.136325
$317$	−20.7685	−1.16647	−0.583237	−	0.812302i	$-0.698214\pi$
$317$	−20.7685	−1.16647	−0.583237	+	0.812302i	$0.698214\pi$
$318$	0	0
$319$	3.63037	0.203261
$320$	0	0
$321$	0	0
$322$	11.4525	0.638221
$323$	−5.09984	−0.283763
$324$	0	0
$325$	0	0
$326$	42.9713	2.37996
$327$	0	0
$328$	40.5312	2.23796
$329$	−37.7515	−2.08131
$330$	0	0
$331$	−33.9172	−1.86426	−0.932128	−	0.362129i	$-0.882050\pi$
$331$	−33.9172	−1.86426	−0.932128	+	0.362129i	$0.882050\pi$
$332$	70.1595	3.85050
$333$	0	0
$334$	48.5085	2.65427
$335$	0	0
$336$	0	0
$337$	17.7792	0.968494	0.484247	−	0.874931i	$-0.339094\pi$
$337$	17.7792	0.968494	0.484247	+	0.874931i	$0.339094\pi$
$338$	36.1339	1.96542
$339$	0	0
$340$	0	0
$341$	−7.81416	−0.423161
$342$	0	0
$343$	12.8441	0.693518
$344$	101.211	5.45693
$345$	0	0
$346$	−53.7181	−2.88790
$347$	12.2251	0.656277	0.328138	−	0.944630i	$-0.393579\pi$
$347$	12.2251	0.656277	0.328138	+	0.944630i	$0.393579\pi$
$348$	0	0
$349$	12.8425	0.687441	0.343720	−	0.939072i	$-0.388313\pi$
$349$	12.8425	0.687441	0.343720	+	0.939072i	$0.388313\pi$
$350$	0	0
$351$	0	0
$352$	83.7579	4.46431
$353$	−17.4163	−0.926976	−0.463488	−	0.886103i	$-0.653402\pi$
$353$	−17.4163	−0.926976	−0.463488	+	0.886103i	$0.653402\pi$
$354$	0	0
$355$	0	0
$356$	87.5749	4.64146
$357$	0	0
$358$	66.3167	3.50494
$359$	3.82947	0.202111	0.101056	−	0.994881i	$-0.467778\pi$
$359$	3.82947	0.202111	0.101056	+	0.994881i	$0.467778\pi$
$360$	0	0
$361$	−15.2325	−0.801712
$362$	29.7918	1.56582
$363$	0	0
$364$	−3.41893	−0.179201
$365$	0	0
$366$	0	0
$367$	−29.6282	−1.54658	−0.773289	−	0.634054i	$-0.781390\pi$
$367$	−29.6282	−1.54658	−0.773289	+	0.634054i	$0.781390\pi$
$368$	23.1336	1.20592
$369$	0	0
$370$	0	0
$371$	1.16399	0.0604314
$372$	0	0
$373$	11.4595	0.593350	0.296675	−	0.954978i	$-0.404122\pi$
$373$	11.4595	0.593350	0.296675	+	0.954978i	$0.404122\pi$
$374$	−21.5794	−1.11584
$375$	0	0
$376$	−125.862	−6.49083
$377$	−0.231820	−0.0119393
$378$	0	0
$379$	−2.19307	−0.112650	−0.0563251	−	0.998412i	$-0.517938\pi$
$379$	−2.19307	−0.112650	−0.0563251	+	0.998412i	$0.517938\pi$
$380$	0	0
$381$	0	0
$382$	−40.0177	−2.04749
$383$	10.1997	0.521181	0.260591	−	0.965449i	$-0.416083\pi$
$383$	10.1997	0.521181	0.260591	+	0.965449i	$0.416083\pi$
$384$	0	0
$385$	0	0
$386$	−5.44904	−0.277349
$387$	0	0
$388$	80.1960	4.07134
$389$	−28.5482	−1.44745	−0.723726	−	0.690088i	$-0.757572\pi$
$389$	−28.5482	−1.44745	−0.723726	+	0.690088i	$0.757572\pi$
$390$	0	0
$391$	−3.42740	−0.173331
$392$	−30.6920	−1.55018
$393$	0	0
$394$	−3.28764	−0.165629
$395$	0	0
$396$	0	0
$397$	37.1827	1.86614	0.933072	−	0.359688i	$-0.117117\pi$
$397$	37.1827	1.86614	0.933072	+	0.359688i	$0.117117\pi$
$398$	18.2785	0.916218
$399$	0	0
$400$	0	0
$401$	−7.38648	−0.368863	−0.184432	−	0.982845i	$-0.559044\pi$
$401$	−7.38648	−0.368863	−0.184432	+	0.982845i	$0.559044\pi$
$402$	0	0
$403$	0.498979	0.0248559
$404$	55.7877	2.77554
$405$	0	0
$406$	−10.8160	−0.536787
$407$	−29.8102	−1.47764
$408$	0	0
$409$	−5.19942	−0.257095	−0.128547	−	0.991703i	$-0.541031\pi$
$409$	−5.19942	−0.257095	−0.128547	+	0.991703i	$0.541031\pi$
$410$	0	0
$411$	0	0
$412$	50.1492	2.47068
$413$	−21.7622	−1.07085
$414$	0	0
$415$	0	0
$416$	−5.34842	−0.262228
$417$	0	0
$418$	15.9416	0.779730
$419$	14.9205	0.728914	0.364457	−	0.931220i	$-0.381255\pi$
$419$	14.9205	0.728914	0.364457	+	0.931220i	$0.381255\pi$
$420$	0	0
$421$	−0.0666925	−0.00325039	−0.00162520	−	0.999999i	$-0.500517\pi$
$421$	−0.0666925	−0.00325039	−0.00162520	+	0.999999i	$0.500517\pi$
$422$	−34.4414	−1.67658
$423$	0	0
$424$	3.88069	0.188463
$425$	0	0
$426$	0	0
$427$	−17.9123	−0.866835
$428$	14.7999	0.715382
$429$	0	0
$430$	0	0
$431$	−27.6996	−1.33424	−0.667121	−	0.744950i	$-0.732473\pi$
$431$	−27.6996	−1.33424	−0.667121	+	0.744950i	$0.732473\pi$
$432$	0	0
$433$	−30.5626	−1.46874	−0.734372	−	0.678748i	$-0.762523\pi$
$433$	−30.5626	−1.46874	−0.734372	+	0.678748i	$0.762523\pi$
$434$	23.2807	1.11751
$435$	0	0
$436$	−87.8770	−4.20854
$437$	2.53197	0.121120
$438$	0	0
$439$	11.0968	0.529623	0.264812	−	0.964300i	$-0.414690\pi$
$439$	11.0968	0.529623	0.264812	+	0.964300i	$0.414690\pi$
$440$	0	0
$441$	0	0
$442$	1.37797	0.0655432
$443$	9.82831	0.466957	0.233479	−	0.972362i	$-0.424989\pi$
$443$	9.82831	0.466957	0.233479	+	0.972362i	$0.424989\pi$
$444$	0	0
$445$	0	0
$446$	20.3653	0.964324
$447$	0	0
$448$	−137.815	−6.51114
$449$	19.1301	0.902805	0.451402	−	0.892320i	$-0.350924\pi$
$449$	19.1301	0.902805	0.451402	+	0.892320i	$0.350924\pi$
$450$	0	0
$451$	11.3729	0.535529
$452$	−12.7122	−0.597933
$453$	0	0
$454$	−58.5858	−2.74957
$455$	0	0
$456$	0	0
$457$	18.7578	0.877452	0.438726	−	0.898621i	$-0.355430\pi$
$457$	18.7578	0.877452	0.438726	+	0.898621i	$0.355430\pi$
$458$	−54.5003	−2.54663
$459$	0	0
$460$	0	0
$461$	37.6208	1.75217	0.876087	−	0.482153i	$-0.160145\pi$
$461$	37.6208	1.75217	0.876087	+	0.482153i	$0.160145\pi$
$462$	0	0
$463$	−19.9275	−0.926111	−0.463056	−	0.886329i	$-0.653247\pi$
$463$	−19.9275	−0.926111	−0.463056	+	0.886329i	$0.653247\pi$
$464$	−21.8479	−1.01426
$465$	0	0
$466$	33.8558	1.56834
$467$	−27.3667	−1.26638	−0.633190	−	0.773997i	$-0.718255\pi$
$467$	−27.3667	−1.26638	−0.633190	+	0.773997i	$0.718255\pi$
$468$	0	0
$469$	−10.2119	−0.471542
$470$	0	0
$471$	0	0
$472$	−72.5542	−3.33958
$473$	28.3994	1.30581
$474$	0	0
$475$	0	0
$476$	47.7386	2.18810
$477$	0	0
$478$	37.4915	1.71482
$479$	38.5742	1.76250	0.881250	−	0.472650i	$-0.156703\pi$
$479$	38.5742	1.76250	0.881250	+	0.472650i	$0.156703\pi$
$480$	0	0
$481$	1.90356	0.0867946
$482$	−55.5669	−2.53100
$483$	0	0
$484$	−13.3607	−0.607303
$485$	0	0
$486$	0	0
$487$	−18.4995	−0.838294	−0.419147	−	0.907918i	$-0.637671\pi$
$487$	−18.4995	−0.838294	−0.419147	+	0.907918i	$0.637671\pi$
$488$	−59.7187	−2.70334
$489$	0	0
$490$	0	0
$491$	−19.7211	−0.890000	−0.445000	−	0.895531i	$-0.646796\pi$
$491$	−19.7211	−0.890000	−0.445000	+	0.895531i	$0.646796\pi$
$492$	0	0
$493$	3.23691	0.145783
$494$	−1.01796	−0.0458004
$495$	0	0
$496$	47.0263	2.11154
$497$	21.1044	0.946663
$498$	0	0
$499$	−36.2906	−1.62459	−0.812296	−	0.583245i	$-0.801783\pi$
$499$	−36.2906	−1.62459	−0.812296	+	0.583245i	$0.801783\pi$
$500$	0	0
$501$	0	0
$502$	−7.69280	−0.343347
$503$	−2.66163	−0.118676	−0.0593382	−	0.998238i	$-0.518899\pi$
$503$	−2.66163	−0.118676	−0.0593382	+	0.998238i	$0.518899\pi$
$504$	0	0
$505$	0	0
$506$	10.7137	0.476283
$507$	0	0
$508$	−86.3410	−3.83076
$509$	−22.8322	−1.01202	−0.506010	−	0.862528i	$-0.668880\pi$
$509$	−22.8322	−1.01202	−0.506010	+	0.862528i	$0.668880\pi$
$510$	0	0
$511$	29.8285	1.31954
$512$	−131.576	−5.81488
$513$	0	0
$514$	−48.1748	−2.12490
$515$	0	0
$516$	0	0
$517$	−35.3163	−1.55321
$518$	88.8137	3.90225
$519$	0	0
$520$	0	0
$521$	12.3029	0.539001	0.269501	−	0.963000i	$-0.413141\pi$
$521$	12.3029	0.539001	0.269501	+	0.963000i	$0.413141\pi$
$522$	0	0
$523$	−33.3902	−1.46005	−0.730025	−	0.683420i	$-0.760492\pi$
$523$	−33.3902	−1.46005	−0.730025	+	0.683420i	$0.760492\pi$
$524$	−65.1541	−2.84627
$525$	0	0
$526$	−11.0531	−0.481939
$527$	−6.96726	−0.303499
$528$	0	0
$529$	−21.2984	−0.926016
$530$	0	0
$531$	0	0
$532$	−35.2666	−1.52900
$533$	−0.726225	−0.0314563
$534$	0	0
$535$	0	0
$536$	−34.0461	−1.47057
$537$	0	0
$538$	3.33238	0.143669
$539$	−8.61204	−0.370947
$540$	0	0
$541$	5.98673	0.257389	0.128695	−	0.991684i	$-0.458921\pi$
$541$	5.98673	0.257389	0.128695	+	0.991684i	$0.458921\pi$
$542$	6.08413	0.261336
$543$	0	0
$544$	74.6802	3.20189
$545$	0	0
$546$	0	0
$547$	4.47889	0.191503	0.0957516	−	0.995405i	$-0.469475\pi$
$547$	4.47889	0.191503	0.0957516	+	0.995405i	$0.469475\pi$
$548$	−24.1033	−1.02964
$549$	0	0
$550$	0	0
$551$	−2.39124	−0.101870
$552$	0	0
$553$	−1.32343	−0.0562780
$554$	−5.43972	−0.231112
$555$	0	0
$556$	−88.5695	−3.75618
$557$	14.4278	0.611326	0.305663	−	0.952140i	$-0.401122\pi$
$557$	14.4278	0.611326	0.305663	+	0.952140i	$0.401122\pi$
$558$	0	0
$559$	−1.81346	−0.0767014
$560$	0	0
$561$	0	0
$562$	6.41401	0.270559
$563$	−27.6973	−1.16730	−0.583652	−	0.812004i	$-0.698377\pi$
$563$	−27.6973	−1.16730	−0.583652	+	0.812004i	$0.698377\pi$
$564$	0	0
$565$	0	0
$566$	29.6715	1.24719
$567$	0	0
$568$	70.3612	2.95229
$569$	−35.6179	−1.49318	−0.746590	−	0.665285i	$-0.768310\pi$
$569$	−35.6179	−1.49318	−0.746590	+	0.665285i	$0.768310\pi$
$570$	0	0
$571$	5.85044	0.244833	0.122417	−	0.992479i	$-0.460936\pi$
$571$	5.85044	0.244833	0.122417	+	0.992479i	$0.460936\pi$
$572$	−3.19839	−0.133731
$573$	0	0
$574$	−33.8833	−1.41426
$575$	0	0
$576$	0	0
$577$	12.0524	0.501746	0.250873	−	0.968020i	$-0.419282\pi$
$577$	12.0524	0.501746	0.250873	+	0.968020i	$0.419282\pi$
$578$	28.1404	1.17049
$579$	0	0
$580$	0	0
$581$	−38.3150	−1.58958
$582$	0	0
$583$	1.08891	0.0450979
$584$	99.4470	4.11515
$585$	0	0
$586$	41.6928	1.72232
$587$	−40.2350	−1.66068	−0.830339	−	0.557259i	$-0.811853\pi$
$587$	−40.2350	−1.66068	−0.830339	+	0.557259i	$0.811853\pi$
$588$	0	0
$589$	5.14702	0.212079
$590$	0	0
$591$	0	0
$592$	179.401	7.37332
$593$	−33.4086	−1.37193	−0.685964	−	0.727636i	$-0.740619\pi$
$593$	−33.4086	−1.37193	−0.685964	+	0.727636i	$0.740619\pi$
$594$	0	0
$595$	0	0
$596$	−19.8053	−0.811258
$597$	0	0
$598$	−0.684132	−0.0279763
$599$	24.6421	1.00685	0.503424	−	0.864040i	$-0.332073\pi$
$599$	24.6421	1.00685	0.503424	+	0.864040i	$0.332073\pi$
$600$	0	0
$601$	16.6327	0.678461	0.339230	−	0.940703i	$-0.389833\pi$
$601$	16.6327	0.678461	0.339230	+	0.940703i	$0.389833\pi$
$602$	−84.6103	−3.44846
$603$	0	0
$604$	62.1310	2.52807
$605$	0	0
$606$	0	0
$607$	29.4057	1.19354	0.596770	−	0.802412i	$-0.296450\pi$
$607$	29.4057	1.19354	0.596770	+	0.802412i	$0.296450\pi$
$608$	−55.1695	−2.23742
$609$	0	0
$610$	0	0
$611$	2.25515	0.0912337
$612$	0	0
$613$	12.1702	0.491549	0.245775	−	0.969327i	$-0.420958\pi$
$613$	12.1702	0.491549	0.245775	+	0.969327i	$0.420958\pi$
$614$	8.56574	0.345685
$615$	0	0
$616$	−97.4838	−3.92774
$617$	39.1024	1.57420	0.787102	−	0.616823i	$-0.211581\pi$
$617$	39.1024	1.57420	0.787102	+	0.616823i	$0.211581\pi$
$618$	0	0
$619$	−4.72717	−0.190001	−0.0950006	−	0.995477i	$-0.530285\pi$
$619$	−4.72717	−0.190001	−0.0950006	+	0.995477i	$0.530285\pi$
$620$	0	0
$621$	0	0
$622$	69.4258	2.78372
$623$	−47.8258	−1.91610
$624$	0	0
$625$	0	0
$626$	−25.8801	−1.03438
$627$	0	0
$628$	−77.3179	−3.08532
$629$	−26.5794	−1.05979
$630$	0	0
$631$	−33.8710	−1.34838	−0.674191	−	0.738557i	$-0.735508\pi$
$631$	−33.8710	−1.34838	−0.674191	+	0.738557i	$0.735508\pi$
$632$	−4.41226	−0.175510
$633$	0	0
$634$	57.8842	2.29888
$635$	0	0
$636$	0	0
$637$	0.549929	0.0217890
$638$	−10.1183	−0.400586
$639$	0	0
$640$	0	0
$641$	−15.9571	−0.630269	−0.315135	−	0.949047i	$-0.602050\pi$
$641$	−15.9571	−0.630269	−0.315135	+	0.949047i	$0.602050\pi$
$642$	0	0
$643$	−10.2436	−0.403968	−0.201984	−	0.979389i	$-0.564739\pi$
$643$	−10.2436	−0.403968	−0.201984	+	0.979389i	$0.564739\pi$
$644$	−23.7013	−0.933960
$645$	0	0
$646$	14.2139	0.559237
$647$	6.00244	0.235980	0.117990	−	0.993015i	$-0.462355\pi$
$647$	6.00244	0.235980	0.117990	+	0.993015i	$0.462355\pi$
$648$	0	0
$649$	−20.3584	−0.799138
$650$	0	0
$651$	0	0
$652$	−88.9305	−3.48279
$653$	−33.1069	−1.29557	−0.647787	−	0.761822i	$-0.724305\pi$
$653$	−33.1069	−1.29557	−0.647787	+	0.761822i	$0.724305\pi$
$654$	0	0
$655$	0	0
$656$	−68.4431	−2.67226
$657$	0	0
$658$	105.218	4.10183
$659$	−19.5593	−0.761922	−0.380961	−	0.924591i	$-0.624407\pi$
$659$	−19.5593	−0.761922	−0.380961	+	0.924591i	$0.624407\pi$
$660$	0	0
$661$	30.5290	1.18744	0.593721	−	0.804671i	$-0.297658\pi$
$661$	30.5290	1.18744	0.593721	+	0.804671i	$0.297658\pi$
$662$	94.5312	3.67406
$663$	0	0
$664$	−127.741	−4.95730
$665$	0	0
$666$	0	0
$667$	−1.60706	−0.0622255
$668$	−100.390	−3.88421
$669$	0	0
$670$	0	0
$671$	−16.7568	−0.646890
$672$	0	0
$673$	4.19687	0.161778	0.0808888	−	0.996723i	$-0.474224\pi$
$673$	4.19687	0.161778	0.0808888	+	0.996723i	$0.474224\pi$
$674$	−49.5527	−1.90870
$675$	0	0
$676$	−74.7802	−2.87616
$677$	−9.66481	−0.371449	−0.185724	−	0.982602i	$-0.559463\pi$
$677$	−9.66481	−0.371449	−0.185724	+	0.982602i	$0.559463\pi$
$678$	0	0
$679$	−43.7961	−1.68074
$680$	0	0
$681$	0	0
$682$	21.7790	0.833961
$683$	−9.03781	−0.345822	−0.172911	−	0.984937i	$-0.555317\pi$
$683$	−9.03781	−0.345822	−0.172911	+	0.984937i	$0.555317\pi$
$684$	0	0
$685$	0	0
$686$	−35.7981	−1.36678
$687$	0	0
$688$	−170.910	−6.51588
$689$	−0.0695329	−0.00264899
$690$	0	0
$691$	−32.1359	−1.22251	−0.611254	−	0.791435i	$-0.709334\pi$
$691$	−32.1359	−1.22251	−0.611254	+	0.791435i	$0.709334\pi$
$692$	111.171	4.22610
$693$	0	0
$694$	−34.0728	−1.29338
$695$	0	0
$696$	0	0
$697$	10.1403	0.384091
$698$	−35.7935	−1.35480
$699$	0	0
$700$	0	0
$701$	8.54116	0.322595	0.161298	−	0.986906i	$-0.448432\pi$
$701$	8.54116	0.322595	0.161298	+	0.986906i	$0.448432\pi$
$702$	0	0
$703$	19.6354	0.740562
$704$	−128.925	−4.85904
$705$	0	0
$706$	48.5413	1.82688
$707$	−30.4664	−1.14581
$708$	0	0
$709$	5.04613	0.189511	0.0947556	−	0.995501i	$-0.469793\pi$
$709$	5.04613	0.189511	0.0947556	+	0.995501i	$0.469793\pi$
$710$	0	0
$711$	0	0
$712$	−159.449	−5.97561
$713$	3.45910	0.129544
$714$	0	0
$715$	0	0
$716$	−137.244	−5.12907
$717$	0	0
$718$	−10.6732	−0.398319
$719$	19.5253	0.728170	0.364085	−	0.931366i	$-0.381382\pi$
$719$	19.5253	0.728170	0.364085	+	0.931366i	$0.381382\pi$
$720$	0	0
$721$	−27.3872	−1.01995
$722$	42.4549	1.58001
$723$	0	0
$724$	−61.6552	−2.29140
$725$	0	0
$726$	0	0
$727$	40.4464	1.50007	0.750037	−	0.661395i	$-0.230035\pi$
$727$	40.4464	1.50007	0.750037	+	0.661395i	$0.230035\pi$
$728$	6.22490	0.230710
$729$	0	0
$730$	0	0
$731$	25.3215	0.936548
$732$	0	0
$733$	29.3862	1.08540	0.542702	−	0.839926i	$-0.317401\pi$
$733$	29.3862	1.08540	0.542702	+	0.839926i	$0.317401\pi$
$734$	82.5772	3.04798
$735$	0	0
$736$	−37.0772	−1.36668
$737$	−9.55318	−0.351896
$738$	0	0
$739$	−30.7254	−1.13025	−0.565126	−	0.825005i	$-0.691172\pi$
$739$	−30.7254	−1.13025	−0.565126	+	0.825005i	$0.691172\pi$
$740$	0	0
$741$	0	0
$742$	−3.24418	−0.119098
$743$	−41.4841	−1.52190	−0.760952	−	0.648808i	$-0.775268\pi$
$743$	−41.4841	−1.52190	−0.760952	+	0.648808i	$0.775268\pi$
$744$	0	0
$745$	0	0
$746$	−31.9390	−1.16937
$747$	0	0
$748$	44.6592	1.63290
$749$	−8.08244	−0.295326
$750$	0	0
$751$	38.6123	1.40898	0.704491	−	0.709713i	$-0.251175\pi$
$751$	38.6123	1.40898	0.704491	+	0.709713i	$0.251175\pi$
$752$	212.537	7.75042
$753$	0	0
$754$	0.646109	0.0235299
$755$	0	0
$756$	0	0
$757$	−23.0860	−0.839074	−0.419537	−	0.907738i	$-0.637807\pi$
$757$	−23.0860	−0.839074	−0.419537	+	0.907738i	$0.637807\pi$
$758$	6.11234	0.222010
$759$	0	0
$760$	0	0
$761$	31.7092	1.14946	0.574728	−	0.818344i	$-0.305108\pi$
$761$	31.7092	1.14946	0.574728	+	0.818344i	$0.305108\pi$
$762$	0	0
$763$	47.9908	1.73738
$764$	82.8180	2.99625
$765$	0	0
$766$	−28.4278	−1.02714
$767$	1.30000	0.0469404
$768$	0	0
$769$	38.6475	1.39366	0.696832	−	0.717235i	$-0.254592\pi$
$769$	38.6475	1.39366	0.696832	+	0.717235i	$0.254592\pi$
$770$	0	0
$771$	0	0
$772$	11.2770	0.405867
$773$	−38.9958	−1.40258	−0.701291	−	0.712875i	$-0.747393\pi$
$773$	−38.9958	−1.40258	−0.701291	+	0.712875i	$0.747393\pi$
$774$	0	0
$775$	0	0
$776$	−146.014	−5.24161
$777$	0	0
$778$	79.5672	2.85262
$779$	−7.49109	−0.268396
$780$	0	0
$781$	19.7431	0.706463
$782$	9.55256	0.341599
$783$	0	0
$784$	51.8281	1.85100
$785$	0	0
$786$	0	0
$787$	−8.11107	−0.289128	−0.144564	−	0.989495i	$-0.546178\pi$
$787$	−8.11107	−0.289128	−0.144564	+	0.989495i	$0.546178\pi$
$788$	6.80387	0.242378
$789$	0	0
$790$	0	0
$791$	6.94231	0.246840
$792$	0	0
$793$	1.07002	0.0379975
$794$	−103.633	−3.67778
$795$	0	0
$796$	−37.8279	−1.34078
$797$	19.0152	0.673554	0.336777	−	0.941584i	$-0.390663\pi$
$797$	19.0152	0.673554	0.336777	+	0.941584i	$0.390663\pi$
$798$	0	0
$799$	−31.4888	−1.11399
$800$	0	0
$801$	0	0
$802$	20.5870	0.726952
$803$	27.9044	0.984726
$804$	0	0
$805$	0	0
$806$	−1.39071	−0.0489858
$807$	0	0
$808$	−101.574	−3.57335
$809$	−49.9078	−1.75467	−0.877333	−	0.479882i	$-0.840679\pi$
$809$	−49.9078	−1.75467	−0.877333	+	0.479882i	$0.840679\pi$
$810$	0	0
$811$	30.9727	1.08760	0.543799	−	0.839215i	$-0.316985\pi$
$811$	30.9727	1.08760	0.543799	+	0.839215i	$0.316985\pi$
$812$	22.3840	0.785523
$813$	0	0
$814$	83.0847	2.91212
$815$	0	0
$816$	0	0
$817$	−18.7061	−0.654443
$818$	14.4914	0.506680
$819$	0	0
$820$	0	0
$821$	−18.9246	−0.660473	−0.330236	−	0.943898i	$-0.607128\pi$
$821$	−18.9246	−0.660473	−0.330236	+	0.943898i	$0.607128\pi$
$822$	0	0
$823$	−9.31122	−0.324569	−0.162284	−	0.986744i	$-0.551886\pi$
$823$	−9.31122	−0.324569	−0.162284	+	0.986744i	$0.551886\pi$
$824$	−91.3076	−3.18085
$825$	0	0
$826$	60.6539	2.11042
$827$	−23.7383	−0.825461	−0.412731	−	0.910853i	$-0.635425\pi$
$827$	−23.7383	−0.825461	−0.412731	+	0.910853i	$0.635425\pi$
$828$	0	0
$829$	1.91351	0.0664590	0.0332295	−	0.999448i	$-0.489421\pi$
$829$	1.91351	0.0664590	0.0332295	+	0.999448i	$0.489421\pi$
$830$	0	0
$831$	0	0
$832$	8.23260	0.285414
$833$	−7.67867	−0.266050
$834$	0	0
$835$	0	0
$836$	−32.9917	−1.14104
$837$	0	0
$838$	−41.5852	−1.43654
$839$	−40.5105	−1.39858	−0.699289	−	0.714839i	$-0.746500\pi$
$839$	−40.5105	−1.39858	−0.699289	+	0.714839i	$0.746500\pi$
$840$	0	0
$841$	−27.4823	−0.947664
$842$	0.185880	0.00640584
$843$	0	0
$844$	71.2776	2.45348
$845$	0	0
$846$	0	0
$847$	7.29643	0.250708
$848$	−6.55314	−0.225036
$849$	0	0
$850$	0	0
$851$	13.1961	0.452358
$852$	0	0
$853$	18.7735	0.642792	0.321396	−	0.946945i	$-0.395848\pi$
$853$	18.7735	0.642792	0.321396	+	0.946945i	$0.395848\pi$
$854$	49.9236	1.70835
$855$	0	0
$856$	−26.9465	−0.921012
$857$	10.0276	0.342536	0.171268	−	0.985225i	$-0.445214\pi$
$857$	10.0276	0.342536	0.171268	+	0.985225i	$0.445214\pi$
$858$	0	0
$859$	−28.0443	−0.956858	−0.478429	−	0.878126i	$-0.658794\pi$
$859$	−28.0443	−0.956858	−0.478429	+	0.878126i	$0.658794\pi$
$860$	0	0
$861$	0	0
$862$	77.2020	2.62951
$863$	−9.84666	−0.335184	−0.167592	−	0.985856i	$-0.553599\pi$
$863$	−9.84666	−0.335184	−0.167592	+	0.985856i	$0.553599\pi$
$864$	0	0
$865$	0	0
$866$	85.1815	2.89459
$867$	0	0
$868$	−48.1802	−1.63534
$869$	−1.23806	−0.0419984
$870$	0	0
$871$	0.610026	0.0206699
$872$	159.999	5.41826
$873$	0	0
$874$	−7.05690	−0.238703
$875$	0	0
$876$	0	0
$877$	36.9671	1.24829	0.624146	−	0.781308i	$-0.285447\pi$
$877$	36.9671	1.24829	0.624146	+	0.781308i	$0.285447\pi$
$878$	−30.9282	−1.04378
$879$	0	0
$880$	0	0
$881$	20.7137	0.697863	0.348932	−	0.937148i	$-0.386545\pi$
$881$	20.7137	0.697863	0.348932	+	0.937148i	$0.386545\pi$
$882$	0	0
$883$	12.7254	0.428243	0.214121	−	0.976807i	$-0.431311\pi$
$883$	12.7254	0.428243	0.214121	+	0.976807i	$0.431311\pi$
$884$	−2.85175	−0.0959146
$885$	0	0
$886$	−27.3927	−0.920275
$887$	21.6903	0.728287	0.364144	−	0.931343i	$-0.381362\pi$
$887$	21.6903	0.728287	0.364144	+	0.931343i	$0.381362\pi$
$888$	0	0
$889$	47.1520	1.58143
$890$	0	0
$891$	0	0
$892$	−42.1466	−1.41117
$893$	23.2621	0.778437
$894$	0	0
$895$	0	0
$896$	205.040	6.84990
$897$	0	0
$898$	−53.3178	−1.77924
$899$	−3.26685	−0.108956
$900$	0	0
$901$	0.970891	0.0323451
$902$	−31.6976	−1.05542
$903$	0	0
$904$	23.1454	0.769803
$905$	0	0
$906$	0	0
$907$	−3.28462	−0.109064	−0.0545320	−	0.998512i	$-0.517367\pi$
$907$	−3.28462	−0.109064	−0.0545320	+	0.998512i	$0.517367\pi$
$908$	121.245	4.02366
$909$	0	0
$910$	0	0
$911$	−28.8264	−0.955060	−0.477530	−	0.878615i	$-0.658468\pi$
$911$	−28.8264	−0.955060	−0.477530	+	0.878615i	$0.658468\pi$
$912$	0	0
$913$	−35.8435	−1.18625
$914$	−52.2802	−1.72927
$915$	0	0
$916$	112.790	3.72669
$917$	35.5815	1.17500
$918$	0	0
$919$	−17.3985	−0.573925	−0.286962	−	0.957942i	$-0.592645\pi$
$919$	−17.3985	−0.573925	−0.286962	+	0.957942i	$0.592645\pi$
$920$	0	0
$921$	0	0
$922$	−104.854	−3.45317
$923$	−1.26071	−0.0414967
$924$	0	0
$925$	0	0
$926$	55.5404	1.82517
$927$	0	0
$928$	35.0165	1.14947
$929$	24.3715	0.799603	0.399801	−	0.916602i	$-0.369079\pi$
$929$	24.3715	0.799603	0.399801	+	0.916602i	$0.369079\pi$
$930$	0	0
$931$	5.67257	0.185911
$932$	−70.0657	−2.29508
$933$	0	0
$934$	76.2742	2.49577
$935$	0	0
$936$	0	0
$937$	14.5784	0.476257	0.238129	−	0.971234i	$-0.423466\pi$
$937$	14.5784	0.476257	0.238129	+	0.971234i	$0.423466\pi$
$938$	28.4618	0.929311
$939$	0	0
$940$	0	0
$941$	4.19686	0.136814	0.0684068	−	0.997658i	$-0.478208\pi$
$941$	4.19686	0.136814	0.0684068	+	0.997658i	$0.478208\pi$
$942$	0	0
$943$	−5.03445	−0.163944
$944$	122.519	3.98765
$945$	0	0
$946$	−79.1525	−2.57347
$947$	5.53251	0.179782	0.0898912	−	0.995952i	$-0.471348\pi$
$947$	5.53251	0.179782	0.0898912	+	0.995952i	$0.471348\pi$
$948$	0	0
$949$	−1.78186	−0.0578416
$950$	0	0
$951$	0	0
$952$	−86.9185	−2.81705
$953$	−50.4779	−1.63514	−0.817570	−	0.575830i	$-0.804679\pi$
$953$	−50.4779	−1.63514	−0.817570	+	0.575830i	$0.804679\pi$
$954$	0	0
$955$	0	0
$956$	−77.5899	−2.50944
$957$	0	0
$958$	−107.511	−3.47352
$959$	13.1631	0.425060
$960$	0	0
$961$	−23.9683	−0.773170
$962$	−5.30544	−0.171054
$963$	0	0
$964$	114.998	3.70382
$965$	0	0
$966$	0	0
$967$	−17.0340	−0.547777	−0.273889	−	0.961761i	$-0.588310\pi$
$967$	−17.0340	−0.547777	−0.273889	+	0.961761i	$0.588310\pi$
$968$	24.3260	0.781867
$969$	0	0
$970$	0	0
$971$	−23.4172	−0.751494	−0.375747	−	0.926722i	$-0.622614\pi$
$971$	−23.4172	−0.751494	−0.375747	+	0.926722i	$0.622614\pi$
$972$	0	0
$973$	48.3690	1.55064
$974$	51.5604	1.65210
$975$	0	0
$976$	100.844	3.22794
$977$	20.7626	0.664253	0.332127	−	0.943235i	$-0.392234\pi$
$977$	20.7626	0.664253	0.332127	+	0.943235i	$0.392234\pi$
$978$	0	0
$979$	−44.7408	−1.42992
$980$	0	0
$981$	0	0
$982$	54.9650	1.75400
$983$	−42.1977	−1.34590	−0.672948	−	0.739690i	$-0.734972\pi$
$983$	−42.1977	−1.34590	−0.672948	+	0.739690i	$0.734972\pi$
$984$	0	0
$985$	0	0
$986$	−9.02164	−0.287308
$987$	0	0
$988$	2.10671	0.0670234
$989$	−12.5716	−0.399753
$990$	0	0
$991$	−3.83926	−0.121958	−0.0609791	−	0.998139i	$-0.519422\pi$
$991$	−3.83926	−0.121958	−0.0609791	+	0.998139i	$0.519422\pi$
$992$	−75.3711	−2.39303
$993$	0	0
$994$	−58.8206	−1.86567
$995$	0	0
$996$	0	0
$997$	42.2091	1.33678	0.668389	−	0.743812i	$-0.266984\pi$
$997$	42.2091	1.33678	0.668389	+	0.743812i	$0.266984\pi$
$998$	101.146	3.20173
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.o.1.1		6
3.2	odd	2		1875.2.a.l.1.6	yes	6
5.4	even	2		5625.2.a.r.1.6		6
15.2	even	4		1875.2.b.e.1249.12		12
15.8	even	4		1875.2.b.e.1249.1		12
15.14	odd	2		1875.2.a.i.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.i.1.1	✓	6	15.14	odd	2
1875.2.a.l.1.6	yes	6	3.2	odd	2
1875.2.b.e.1249.1		12	15.8	even	4
1875.2.b.e.1249.12		12	15.2	even	4
5625.2.a.o.1.1		6	1.1	even	1	trivial
5625.2.a.r.1.6		6	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q-2.78712 q^{2} +5.76803 q^{4} -3.15000 q^{7} -10.5020 q^{8} +O(q^{10})\)	\(q-2.78712 q^{2} +5.76803 q^{4} -3.15000 q^{7} -10.5020 q^{8} -2.94681 q^{11} +0.188171 q^{13} +8.77943 q^{14} +17.7341 q^{16} -2.62743 q^{17} +1.94100 q^{19} +8.21310 q^{22} +1.30447 q^{23} -0.524454 q^{26} -18.1693 q^{28} -1.23197 q^{29} +2.65174 q^{31} -28.4233 q^{32} +7.32297 q^{34} +10.1161 q^{37} -5.40980 q^{38} -3.85940 q^{41} -9.63734 q^{43} -16.9973 q^{44} -3.63570 q^{46} +11.9846 q^{47} +2.92250 q^{49} +1.08537 q^{52} -0.369521 q^{53} +33.0812 q^{56} +3.43364 q^{58} +6.90864 q^{59} +5.68643 q^{61} -7.39071 q^{62} +43.7507 q^{64} +3.24188 q^{67} -15.1551 q^{68} -6.69982 q^{71} -9.46938 q^{73} -28.1948 q^{74} +11.1957 q^{76} +9.28244 q^{77} +0.420137 q^{79} +10.7566 q^{82} +12.1635 q^{83} +26.8604 q^{86} +30.9473 q^{88} +15.1828 q^{89} -0.592737 q^{91} +7.52421 q^{92} -33.4025 q^{94} +13.9035 q^{97} -8.14536 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8	\( 6 q - q^{2} + 11 q^{4} - 2 q^{7} - 6 q^{8} + 4 q^{14} + 17 q^{16} - 2 q^{17} - 2 q^{19} + 9 q^{22} + q^{23} - 37 q^{26} - 44 q^{28} - 31 q^{29} - 2 q^{31} - 33 q^{32} + 37 q^{34} - 22 q^{37} - 27 q^{38} - 33 q^{41} - 3 q^{43} + 11 q^{44} - 12 q^{46} + 6 q^{47} + 4 q^{49} - 33 q^{52} - 14 q^{53} + 30 q^{56} - q^{58} + 8 q^{59} + 34 q^{61} - 31 q^{62} + 12 q^{64} + 2 q^{67} - 27 q^{68} + 3 q^{71} - 36 q^{73} - 36 q^{74} + 27 q^{76} - 16 q^{77} + 25 q^{79} + 36 q^{82} + 12 q^{83} + 30 q^{86} + 56 q^{88} - 18 q^{89} + 28 q^{91} - 3 q^{92} - 50 q^{94} + 7 q^{97} - 15 q^{98}+O(q^{100}) \) 6 * q - q^2 + 11 * q^4 - 2 * q^7 - 6 * q^8 + 4 * q^14 + 17 * q^16 - 2 * q^17 - 2 * q^19 + 9 * q^22 + q^23 - 37 * q^26 - 44 * q^28 - 31 * q^29 - 2 * q^31 - 33 * q^32 + 37 * q^34 - 22 * q^37 - 27 * q^38 - 33 * q^41 - 3 * q^43 + 11 * q^44 - 12 * q^46 + 6 * q^47 + 4 * q^49 - 33 * q^52 - 14 * q^53 + 30 * q^56 - q^58 + 8 * q^59 + 34 * q^61 - 31 * q^62 + 12 * q^64 + 2 * q^67 - 27 * q^68 + 3 * q^71 - 36 * q^73 - 36 * q^74 + 27 * q^76 - 16 * q^77 + 25 * q^79 + 36 * q^82 + 12 * q^83 + 30 * q^86 + 56 * q^88 - 18 * q^89 + 28 * q^91 - 3 * q^92 - 50 * q^94 + 7 * q^97 - 15 * q^98

Introduction

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