LMFDB - Embedded newform 3150.3.c.d.449.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,3,Mod(449,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("3150.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3150.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:	$85.8312832735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.9671731157401600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 7x^{12} + 753x^{8} + 112x^{4} + 256 $ x^16 + 7x^12 + 753x^8 + 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 630)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.5
Root			$0.359610 + 0.796626i$ of defining polynomial
Character	$\chi$	$=$	3150.449
Dual form			3150.3.c.d.449.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})$	$q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +2.32744i q^{11} -10.6912i q^{13} -3.74166i q^{14} +4.00000 q^{16} -2.70514 q^{17} -11.0962 q^{19} -3.29150i q^{22} +34.7065 q^{23} +15.1196i q^{26} +5.29150i q^{28} -5.82252i q^{29} -15.2705 q^{31} -5.65685 q^{32} +3.82564 q^{34} -0.900842i q^{37} +15.6924 q^{38} +31.6661i q^{41} -13.0752i q^{43} +4.65489i q^{44} -49.0825 q^{46} -17.1532 q^{47} -7.00000 q^{49} -21.3823i q^{52} +68.1864 q^{53} -7.48331i q^{56} +8.23429i q^{58} -34.3712i q^{59} -81.5400 q^{61} +21.5958 q^{62} +8.00000 q^{64} -46.2125i q^{67} -5.41027 q^{68} +61.3258i q^{71} +20.5408i q^{73} +1.27398i q^{74} -22.1923 q^{76} -6.15784 q^{77} -78.0870 q^{79} -44.7826i q^{82} -28.7135 q^{83} +18.4911i q^{86} -6.58301i q^{88} -136.885i q^{89} +28.2861 q^{91} +69.4131 q^{92} +24.2582 q^{94} +10.0195i q^{97} +9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} + 64 q^{16} - 160 q^{19} - 224 q^{31} + 64 q^{34} - 64 q^{46} - 112 q^{49} - 160 q^{61} + 128 q^{64} - 320 q^{76} + 128 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 + 64 * q^16 - 160 * q^19 - 224 * q^31 + 64 * q^34 - 64 * q^46 - 112 * q^49 - 160 * q^61 + 128 * q^64 - 320 * q^76 + 128 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	−2.82843	−0.353553
$9$	0	0
$10$	0	0
$11$	2.32744i	0.211586i	0.994388	+	0.105793i	$0.0337381\pi$
$11$	2.32744i	0.211586i	−0.994388	+	0.105793i	$0.966262\pi$
$12$	0	0
$13$	− 10.6912i	− 0.822397i	−0.911546	−	0.411198i	$-0.865110\pi$
$13$	− 10.6912i	− 0.822397i	0.911546	−	0.411198i	$-0.134890\pi$
$14$	− 3.74166i	− 0.267261i
$15$	0	0
$16$	4.00000	0.250000
$17$	−2.70514	−0.159126	−0.0795628	−	0.996830i	$-0.525352\pi$
$17$	−2.70514	−0.159126	−0.0795628	+	0.996830i	$0.525352\pi$
$18$	0	0
$19$	−11.0962	−0.584009	−0.292005	−	0.956417i	$-0.594322\pi$
$19$	−11.0962	−0.584009	−0.292005	+	0.956417i	$0.594322\pi$
$20$	0	0
$21$	0	0
$22$	− 3.29150i	− 0.149614i
$23$	34.7065	1.50898	0.754490	−	0.656312i	$-0.227884\pi$
$23$	34.7065	1.50898	0.754490	+	0.656312i	$0.227884\pi$
$24$	0	0
$25$	0	0
$26$	15.1196i	0.581522i
$27$	0	0
$28$	5.29150i	0.188982i
$29$	− 5.82252i	− 0.200777i	−0.994948	−	0.100388i	$-0.967992\pi$
$29$	− 5.82252i	− 0.200777i	0.994948	−	0.100388i	$-0.0320085\pi$
$30$	0	0
$31$	−15.2705	−0.492598	−0.246299	−	0.969194i	$-0.579214\pi$
$31$	−15.2705	−0.492598	−0.246299	+	0.969194i	$0.579214\pi$
$32$	−5.65685	−0.176777
$33$	0	0
$34$	3.82564	0.112519
$35$	0	0
$36$	0	0
$37$	− 0.900842i	− 0.0243471i	−0.999926	−	0.0121735i	$-0.996125\pi$
$37$	− 0.900842i	− 0.0243471i	0.999926	−	0.0121735i	$-0.00387505\pi$
$38$	15.6924	0.412957
$39$	0	0
$40$	0	0
$41$	31.6661i	0.772344i	0.922427	+	0.386172i	$0.126203\pi$
$41$	31.6661i	0.772344i	−0.922427	+	0.386172i	$0.873797\pi$
$42$	0	0
$43$	− 13.0752i	− 0.304074i	−0.988375	−	0.152037i	$-0.951417\pi$
$43$	− 13.0752i	− 0.304074i	0.988375	−	0.152037i	$-0.0485834\pi$
$44$	4.65489i	0.105793i
$45$	0	0
$46$	−49.0825	−1.06701
$47$	−17.1532	−0.364961	−0.182480	−	0.983209i	$-0.558413\pi$
$47$	−17.1532	−0.364961	−0.182480	+	0.983209i	$0.558413\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 21.3823i	− 0.411198i
$53$	68.1864	1.28654	0.643268	−	0.765641i	$-0.277578\pi$
$53$	68.1864	1.28654	0.643268	+	0.765641i	$0.277578\pi$
$54$	0	0
$55$	0	0
$56$	− 7.48331i	− 0.133631i
$57$	0	0
$58$	8.23429i	0.141970i
$59$	− 34.3712i	− 0.582563i	−0.956637	−	0.291282i	$-0.905918\pi$
$59$	− 34.3712i	− 0.582563i	0.956637	−	0.291282i	$-0.0940816\pi$
$60$	0	0
$61$	−81.5400	−1.33672	−0.668360	−	0.743838i	$-0.733004\pi$
$61$	−81.5400	−1.33672	−0.668360	+	0.743838i	$0.733004\pi$
$62$	21.5958	0.348319
$63$	0	0
$64$	8.00000	0.125000
$65$	0	0
$66$	0	0
$67$	− 46.2125i	− 0.689739i	−0.938651	−	0.344869i	$-0.887923\pi$
$67$	− 46.2125i	− 0.689739i	0.938651	−	0.344869i	$-0.112077\pi$
$68$	−5.41027	−0.0795628
$69$	0	0
$70$	0	0
$71$	61.3258i	0.863744i	0.901935	+	0.431872i	$0.142147\pi$
$71$	61.3258i	0.863744i	−0.901935	+	0.431872i	$0.857853\pi$
$72$	0	0
$73$	20.5408i	0.281380i	0.990054	+	0.140690i	$0.0449321\pi$
$73$	20.5408i	0.281380i	−0.990054	+	0.140690i	$0.955068\pi$
$74$	1.27398i	0.0172160i
$75$	0	0
$76$	−22.1923	−0.292005
$77$	−6.15784	−0.0799719
$78$	0	0
$79$	−78.0870	−0.988443	−0.494222	−	0.869336i	$-0.664547\pi$
$79$	−78.0870	−0.988443	−0.494222	+	0.869336i	$0.664547\pi$
$80$	0	0
$81$	0	0
$82$	− 44.7826i	− 0.546129i
$83$	−28.7135	−0.345945	−0.172973	−	0.984927i	$-0.555337\pi$
$83$	−28.7135	−0.345945	−0.172973	+	0.984927i	$0.555337\pi$
$84$	0	0
$85$	0	0
$86$	18.4911i	0.215013i
$87$	0	0
$88$	− 6.58301i	− 0.0748069i
$89$	− 136.885i	− 1.53804i	−0.639227	−	0.769019i	$-0.720745\pi$
$89$	− 136.885i	− 1.53804i	0.639227	−	0.769019i	$-0.279255\pi$
$90$	0	0
$91$	28.2861	0.310837
$92$	69.4131	0.754490
$93$	0	0
$94$	24.2582	0.258066
$95$	0	0
$96$	0	0
$97$	10.0195i	0.103294i	0.998665	+	0.0516470i	$0.0164471\pi$
$97$	10.0195i	0.103294i	−0.998665	+	0.0516470i	$0.983553\pi$
$98$	9.89949	0.101015
$99$	0	0
$100$	0	0
$101$	− 37.1254i	− 0.367578i	−0.982966	−	0.183789i	$-0.941164\pi$
$101$	− 37.1254i	− 0.367578i	0.982966	−	0.183789i	$-0.0588364\pi$
$102$	0	0
$103$	− 107.296i	− 1.04171i	−0.853645	−	0.520856i	$-0.825613\pi$
$103$	− 107.296i	− 1.04171i	0.853645	−	0.520856i	$-0.174387\pi$
$104$	30.2392i	0.290761i
$105$	0	0
$106$	−96.4301	−0.909718
$107$	142.217	1.32913	0.664564	−	0.747232i	$-0.268617\pi$
$107$	142.217	1.32913	0.664564	+	0.747232i	$0.268617\pi$
$108$	0	0
$109$	47.5626	0.436355	0.218177	−	0.975909i	$-0.429989\pi$
$109$	47.5626	0.436355	0.218177	+	0.975909i	$0.429989\pi$
$110$	0	0
$111$	0	0
$112$	10.5830i	0.0944911i
$113$	−103.302	−0.914173	−0.457087	−	0.889422i	$-0.651107\pi$
$113$	−103.302	−0.914173	−0.457087	+	0.889422i	$0.651107\pi$
$114$	0	0
$115$	0	0
$116$	− 11.6450i	− 0.100388i
$117$	0	0
$118$	48.6082i	0.411934i
$119$	− 7.15712i	− 0.0601438i
$120$	0	0
$121$	115.583	0.955231
$122$	115.315	0.945204
$123$	0	0
$124$	−30.5411	−0.246299
$125$	0	0
$126$	0	0
$127$	− 104.802i	− 0.825216i	−0.910909	−	0.412608i	$-0.864618\pi$
$127$	− 104.802i	− 0.825216i	0.910909	−	0.412608i	$-0.135382\pi$
$128$	−11.3137	−0.0883883
$129$	0	0
$130$	0	0
$131$	168.925i	1.28951i	0.764390	+	0.644754i	$0.223040\pi$
$131$	168.925i	1.28951i	−0.764390	+	0.644754i	$0.776960\pi$
$132$	0	0
$133$	− 29.3577i	− 0.220735i
$134$	65.3543i	0.487719i
$135$	0	0
$136$	7.65128	0.0562594
$137$	110.555	0.806973	0.403486	−	0.914986i	$-0.367798\pi$
$137$	110.555	0.806973	0.403486	+	0.914986i	$0.367798\pi$
$138$	0	0
$139$	−223.453	−1.60758	−0.803788	−	0.594916i	$-0.797185\pi$
$139$	−223.453	−1.60758	−0.803788	+	0.594916i	$0.797185\pi$
$140$	0	0
$141$	0	0
$142$	− 86.7278i	− 0.610759i
$143$	24.8831	0.174007
$144$	0	0
$145$	0	0
$146$	− 29.0490i	− 0.198966i
$147$	0	0
$148$	− 1.80168i	− 0.0121735i
$149$	80.4313i	0.539808i	0.962887	+	0.269904i	$0.0869919\pi$
$149$	80.4313i	0.539808i	−0.962887	+	0.269904i	$0.913008\pi$
$150$	0	0
$151$	−138.258	−0.915613	−0.457807	−	0.889052i	$-0.651365\pi$
$151$	−138.258	−0.915613	−0.457807	+	0.889052i	$0.651365\pi$
$152$	31.3847	0.206478
$153$	0	0
$154$	8.70850	0.0565487
$155$	0	0
$156$	0	0
$157$	1.50532i	0.00958802i	0.999989	+	0.00479401i	$0.00152599\pi$
$157$	1.50532i	0.00958802i	−0.999989	+	0.00479401i	$0.998474\pi$
$158$	110.432	0.698935
$159$	0	0
$160$	0	0
$161$	91.8249i	0.570341i
$162$	0	0
$163$	− 222.982i	− 1.36799i	−0.729488	−	0.683993i	$-0.760242\pi$
$163$	− 222.982i	− 1.36799i	0.729488	−	0.683993i	$-0.239758\pi$
$164$	63.3322i	0.386172i
$165$	0	0
$166$	40.6070	0.244620
$167$	120.483	0.721455	0.360728	−	0.932671i	$-0.382528\pi$
$167$	120.483	0.721455	0.360728	+	0.932671i	$0.382528\pi$
$168$	0	0
$169$	54.6992	0.323664
$170$	0	0
$171$	0	0
$172$	− 26.1504i	− 0.152037i
$173$	100.276	0.579631	0.289816	−	0.957082i	$-0.406406\pi$
$173$	100.276	0.579631	0.289816	+	0.957082i	$0.406406\pi$
$174$	0	0
$175$	0	0
$176$	9.30978i	0.0528965i
$177$	0	0
$178$	193.585i	1.08756i
$179$	− 91.4566i	− 0.510931i	−0.966818	−	0.255465i	$-0.917771\pi$
$179$	− 91.4566i	− 0.510931i	0.966818	−	0.255465i	$-0.0822287\pi$
$180$	0	0
$181$	130.398	0.720429	0.360214	−	0.932870i	$-0.382704\pi$
$181$	130.398	0.720429	0.360214	+	0.932870i	$0.382704\pi$
$182$	−40.0026	−0.219795
$183$	0	0
$184$	−98.1649	−0.533505
$185$	0	0
$186$	0	0
$187$	− 6.29605i	− 0.0336687i
$188$	−34.3063	−0.182480
$189$	0	0
$190$	0	0
$191$	− 118.977i	− 0.622916i	−0.950260	−	0.311458i	$-0.899183\pi$
$191$	− 118.977i	− 0.622916i	0.950260	−	0.311458i	$-0.100817\pi$
$192$	0	0
$193$	− 202.816i	− 1.05086i	−0.850837	−	0.525430i	$-0.823904\pi$
$193$	− 202.816i	− 1.05086i	0.850837	−	0.525430i	$-0.176096\pi$
$194$	− 14.1697i	− 0.0730399i
$195$	0	0
$196$	−14.0000	−0.0714286
$197$	295.867	1.50186	0.750932	−	0.660379i	$-0.229604\pi$
$197$	295.867	1.50186	0.750932	+	0.660379i	$0.229604\pi$
$198$	0	0
$199$	−303.023	−1.52273	−0.761365	−	0.648323i	$-0.775471\pi$
$199$	−303.023	−1.52273	−0.761365	+	0.648323i	$0.775471\pi$
$200$	0	0
$201$	0	0
$202$	52.5033i	0.259917i
$203$	15.4049	0.0758864
$204$	0	0
$205$	0	0
$206$	151.740i	0.736601i
$207$	0	0
$208$	− 42.7646i	− 0.205599i
$209$	− 25.8257i	− 0.123568i
$210$	0	0
$211$	234.073	1.10935	0.554676	−	0.832067i	$-0.312842\pi$
$211$	234.073	1.10935	0.554676	+	0.832067i	$0.312842\pi$
$212$	136.373	0.643268
$213$	0	0
$214$	−201.125	−0.939835
$215$	0	0
$216$	0	0
$217$	− 40.4020i	− 0.186184i
$218$	−67.2637	−0.308549
$219$	0	0
$220$	0	0
$221$	28.9210i	0.130864i
$222$	0	0
$223$	− 301.026i	− 1.34989i	−0.737867	−	0.674946i	$-0.764167\pi$
$223$	− 301.026i	− 1.34989i	0.737867	−	0.674946i	$-0.235833\pi$
$224$	− 14.9666i	− 0.0668153i
$225$	0	0
$226$	146.090	0.646418
$227$	−97.7782	−0.430741	−0.215371	−	0.976532i	$-0.569096\pi$
$227$	−97.7782	−0.430741	−0.215371	+	0.976532i	$0.569096\pi$
$228$	0	0
$229$	−71.7468	−0.313305	−0.156652	−	0.987654i	$-0.550070\pi$
$229$	−71.7468	−0.313305	−0.156652	+	0.987654i	$0.550070\pi$
$230$	0	0
$231$	0	0
$232$	16.4686i	0.0709852i
$233$	−321.396	−1.37938	−0.689691	−	0.724104i	$-0.742253\pi$
$233$	−321.396	−1.37938	−0.689691	+	0.724104i	$0.742253\pi$
$234$	0	0
$235$	0	0
$236$	− 68.7424i	− 0.291282i
$237$	0	0
$238$	10.1217i	0.0425281i
$239$	− 37.4949i	− 0.156882i	−0.996919	−	0.0784411i	$-0.975006\pi$
$239$	− 37.4949i	− 0.156882i	0.996919	−	0.0784411i	$-0.0249943\pi$
$240$	0	0
$241$	−153.379	−0.636427	−0.318213	−	0.948019i	$-0.603083\pi$
$241$	−153.379	−0.636427	−0.318213	+	0.948019i	$0.603083\pi$
$242$	−163.459	−0.675451
$243$	0	0
$244$	−163.080	−0.668360
$245$	0	0
$246$	0	0
$247$	118.631i	0.480287i
$248$	43.1916	0.174160
$249$	0	0
$250$	0	0
$251$	121.515i	0.484122i	0.970261	+	0.242061i	$0.0778234\pi$
$251$	121.515i	0.484122i	−0.970261	+	0.242061i	$0.922177\pi$
$252$	0	0
$253$	80.7775i	0.319279i
$254$	148.213i	0.583516i
$255$	0	0
$256$	16.0000	0.0625000
$257$	387.420	1.50747	0.753736	−	0.657177i	$-0.228250\pi$
$257$	387.420	1.50747	0.753736	+	0.657177i	$0.228250\pi$
$258$	0	0
$259$	2.38340	0.00920233
$260$	0	0
$261$	0	0
$262$	− 238.897i	− 0.911819i
$263$	−229.701	−0.873387	−0.436694	−	0.899610i	$-0.643851\pi$
$263$	−229.701	−0.873387	−0.436694	+	0.899610i	$0.643851\pi$
$264$	0	0
$265$	0	0
$266$	41.5181i	0.156083i
$267$	0	0
$268$	− 92.4250i	− 0.344869i
$269$	− 406.720i	− 1.51197i	−0.654588	−	0.755986i	$-0.727158\pi$
$269$	− 406.720i	− 1.51197i	0.654588	−	0.755986i	$-0.272842\pi$
$270$	0	0
$271$	139.965	0.516474	0.258237	−	0.966082i	$-0.416858\pi$
$271$	139.965	0.516474	0.258237	+	0.966082i	$0.416858\pi$
$272$	−10.8205	−0.0397814
$273$	0	0
$274$	−156.349	−0.570616
$275$	0	0
$276$	0	0
$277$	− 476.481i	− 1.72015i	−0.510170	−	0.860074i	$-0.670417\pi$
$277$	− 476.481i	− 1.72015i	0.510170	−	0.860074i	$-0.329583\pi$
$278$	316.010	1.13673
$279$	0	0
$280$	0	0
$281$	− 292.967i	− 1.04259i	−0.853377	−	0.521294i	$-0.825450\pi$
$281$	− 292.967i	− 1.04259i	0.853377	−	0.521294i	$-0.174550\pi$
$282$	0	0
$283$	157.100i	0.555124i	0.960708	+	0.277562i	$0.0895264\pi$
$283$	157.100i	0.555124i	−0.960708	+	0.277562i	$0.910474\pi$
$284$	122.652i	0.431872i
$285$	0	0
$286$	−35.1900	−0.123042
$287$	−83.7806	−0.291918
$288$	0	0
$289$	−281.682	−0.974679
$290$	0	0
$291$	0	0
$292$	41.0815i	0.140690i
$293$	−30.3825	−0.103695	−0.0518473	−	0.998655i	$-0.516511\pi$
$293$	−30.3825	−0.103695	−0.0518473	+	0.998655i	$0.516511\pi$
$294$	0	0
$295$	0	0
$296$	2.54797i	0.00860799i
$297$	0	0
$298$	− 113.747i	− 0.381702i
$299$	− 371.053i	− 1.24098i
$300$	0	0
$301$	34.5937	0.114929
$302$	195.526	0.647436
$303$	0	0
$304$	−44.3847	−0.146002
$305$	0	0
$306$	0	0
$307$	348.067i	1.13377i	0.823797	+	0.566885i	$0.191851\pi$
$307$	348.067i	1.13377i	−0.823797	+	0.566885i	$0.808149\pi$
$308$	−12.3157	−0.0399860
$309$	0	0
$310$	0	0
$311$	23.3072i	0.0749427i	0.999298	+	0.0374713i	$0.0119303\pi$
$311$	23.3072i	0.0749427i	−0.999298	+	0.0374713i	$0.988070\pi$
$312$	0	0
$313$	543.057i	1.73501i	0.497431	+	0.867504i	$0.334277\pi$
$313$	543.057i	1.73501i	−0.497431	+	0.867504i	$0.665723\pi$
$314$	− 2.12884i	− 0.00677975i
$315$	0	0
$316$	−156.174	−0.494222
$317$	434.349	1.37019	0.685093	−	0.728456i	$-0.259762\pi$
$317$	434.349	1.37019	0.685093	+	0.728456i	$0.259762\pi$
$318$	0	0
$319$	13.5516	0.0424815
$320$	0	0
$321$	0	0
$322$	− 129.860i	− 0.403292i
$323$	30.0167	0.0929308
$324$	0	0
$325$	0	0
$326$	315.344i	0.967313i
$327$	0	0
$328$	− 89.5652i	− 0.273065i
$329$	− 45.3830i	− 0.137942i
$330$	0	0
$331$	88.4517	0.267226	0.133613	−	0.991034i	$-0.457342\pi$
$331$	88.4517	0.267226	0.133613	+	0.991034i	$0.457342\pi$
$332$	−57.4269	−0.172973
$333$	0	0
$334$	−170.389	−0.510146
$335$	0	0
$336$	0	0
$337$	− 421.409i	− 1.25047i	−0.780436	−	0.625235i	$-0.785003\pi$
$337$	− 421.409i	− 1.25047i	0.780436	−	0.625235i	$-0.214997\pi$
$338$	−77.3563	−0.228865
$339$	0	0
$340$	0	0
$341$	− 35.5413i	− 0.104227i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	36.9823i	0.107507i
$345$	0	0
$346$	−141.812	−0.409861
$347$	−86.4624	−0.249171	−0.124586	−	0.992209i	$-0.539760\pi$
$347$	−86.4624	−0.249171	−0.124586	+	0.992209i	$0.539760\pi$
$348$	0	0
$349$	17.8784	0.0512275	0.0256138	−	0.999672i	$-0.491846\pi$
$349$	17.8784	0.0512275	0.0256138	+	0.999672i	$0.491846\pi$
$350$	0	0
$351$	0	0
$352$	− 13.1660i	− 0.0374034i
$353$	118.434	0.335506	0.167753	−	0.985829i	$-0.446349\pi$
$353$	118.434	0.335506	0.167753	+	0.985829i	$0.446349\pi$
$354$	0	0
$355$	0	0
$356$	− 273.771i	− 0.769019i
$357$	0	0
$358$	129.339i	0.361283i
$359$	− 112.800i	− 0.314207i	−0.987582	−	0.157104i	$-0.949784\pi$
$359$	− 112.800i	− 0.314207i	0.987582	−	0.157104i	$-0.0502156\pi$
$360$	0	0
$361$	−237.875	−0.658933
$362$	−184.410	−0.509420
$363$	0	0
$364$	56.5723	0.155418
$365$	0	0
$366$	0	0
$367$	− 604.221i	− 1.64638i	−0.567767	−	0.823189i	$-0.692193\pi$
$367$	− 604.221i	− 1.64638i	0.567767	−	0.823189i	$-0.307807\pi$
$368$	138.826	0.377245
$369$	0	0
$370$	0	0
$371$	180.404i	0.486265i
$372$	0	0
$373$	− 133.363i	− 0.357542i	−0.983891	−	0.178771i	$-0.942788\pi$
$373$	− 133.363i	− 0.357542i	0.983891	−	0.178771i	$-0.0572120\pi$
$374$	8.90396i	0.0238074i
$375$	0	0
$376$	48.5165	0.129033
$377$	−62.2495	−0.165118
$378$	0	0
$379$	278.552	0.734966	0.367483	−	0.930030i	$-0.380220\pi$
$379$	278.552	0.734966	0.367483	+	0.930030i	$0.380220\pi$
$380$	0	0
$381$	0	0
$382$	168.259i	0.440468i
$383$	447.329	1.16796	0.583981	−	0.811767i	$-0.301494\pi$
$383$	447.329	1.16796	0.583981	+	0.811767i	$0.301494\pi$
$384$	0	0
$385$	0	0
$386$	286.825i	0.743071i
$387$	0	0
$388$	20.0390i	0.0516470i
$389$	51.7707i	0.133087i	0.997784	+	0.0665433i	$0.0211971\pi$
$389$	51.7707i	0.133087i	−0.997784	+	0.0665433i	$0.978803\pi$
$390$	0	0
$391$	−93.8859	−0.240117
$392$	19.7990	0.0505076
$393$	0	0
$394$	−418.420	−1.06198
$395$	0	0
$396$	0	0
$397$	− 386.835i	− 0.974395i	−0.873292	−	0.487197i	$-0.838019\pi$
$397$	− 386.835i	− 0.974395i	0.873292	−	0.487197i	$-0.161981\pi$
$398$	428.540	1.07673
$399$	0	0
$400$	0	0
$401$	− 611.141i	− 1.52404i	−0.647552	−	0.762021i	$-0.724207\pi$
$401$	− 611.141i	− 1.52404i	0.647552	−	0.762021i	$-0.275793\pi$
$402$	0	0
$403$	163.260i	0.405111i
$404$	− 74.2508i	− 0.183789i
$405$	0	0
$406$	−21.7859	−0.0536598
$407$	2.09666	0.00515150
$408$	0	0
$409$	320.232	0.782964	0.391482	−	0.920186i	$-0.371962\pi$
$409$	320.232	0.782964	0.391482	+	0.920186i	$0.371962\pi$
$410$	0	0
$411$	0	0
$412$	− 214.592i	− 0.520856i
$413$	90.9377	0.220188
$414$	0	0
$415$	0	0
$416$	60.4783i	0.145381i
$417$	0	0
$418$	36.5231i	0.0873758i
$419$	− 467.521i	− 1.11580i	−0.829908	−	0.557901i	$-0.811607\pi$
$419$	− 467.521i	− 1.11580i	0.829908	−	0.557901i	$-0.188393\pi$
$420$	0	0
$421$	176.598	0.419473	0.209736	−	0.977758i	$-0.432739\pi$
$421$	176.598	0.419473	0.209736	+	0.977758i	$0.432739\pi$
$422$	−331.030	−0.784430
$423$	0	0
$424$	−192.860	−0.454859
$425$	0	0
$426$	0	0
$427$	− 215.734i	− 0.505233i
$428$	284.433	0.664564
$429$	0	0
$430$	0	0
$431$	− 159.316i	− 0.369643i	−0.982772	−	0.184821i	$-0.940829\pi$
$431$	− 159.316i	− 0.369643i	0.982772	−	0.184821i	$-0.0591707\pi$
$432$	0	0
$433$	− 310.841i	− 0.717879i	−0.933361	−	0.358939i	$-0.883139\pi$
$433$	− 310.841i	− 0.717879i	0.933361	−	0.358939i	$-0.116861\pi$
$434$	57.1371i	0.131652i
$435$	0	0
$436$	95.1253	0.218177
$437$	−385.110	−0.881258
$438$	0	0
$439$	802.810	1.82872	0.914362	−	0.404897i	$-0.132693\pi$
$439$	802.810	1.82872	0.914362	+	0.404897i	$0.132693\pi$
$440$	0	0
$441$	0	0
$442$	− 40.9005i	− 0.0925351i
$443$	−656.281	−1.48145	−0.740724	−	0.671810i	$-0.765517\pi$
$443$	−656.281	−1.48145	−0.740724	+	0.671810i	$0.765517\pi$
$444$	0	0
$445$	0	0
$446$	425.715i	0.954518i
$447$	0	0
$448$	21.1660i	0.0472456i
$449$	− 366.837i	− 0.817010i	−0.912756	−	0.408505i	$-0.866050\pi$
$449$	− 366.837i	− 0.817010i	0.912756	−	0.408505i	$-0.133950\pi$
$450$	0	0
$451$	−73.7010	−0.163417
$452$	−206.603	−0.457087
$453$	0	0
$454$	138.279	0.304580
$455$	0	0
$456$	0	0
$457$	− 637.661i	− 1.39532i	−0.716429	−	0.697660i	$-0.754225\pi$
$457$	− 637.661i	− 1.39532i	0.716429	−	0.697660i	$-0.245775\pi$
$458$	101.465	0.221540
$459$	0	0
$460$	0	0
$461$	− 591.806i	− 1.28374i	−0.766811	−	0.641872i	$-0.778158\pi$
$461$	− 591.806i	− 1.28374i	0.766811	−	0.641872i	$-0.221842\pi$
$462$	0	0
$463$	− 899.898i	− 1.94362i	−0.235756	−	0.971812i	$-0.575757\pi$
$463$	− 899.898i	− 1.94362i	0.235756	−	0.971812i	$-0.424243\pi$
$464$	− 23.2901i	− 0.0501941i
$465$	0	0
$466$	454.522	0.975370
$467$	418.700	0.896574	0.448287	−	0.893890i	$-0.352034\pi$
$467$	418.700	0.896574	0.448287	+	0.893890i	$0.352034\pi$
$468$	0	0
$469$	122.267	0.260697
$470$	0	0
$471$	0	0
$472$	97.2165i	0.205967i
$473$	30.4318	0.0643378
$474$	0	0
$475$	0	0
$476$	− 14.3142i	− 0.0300719i
$477$	0	0
$478$	53.0257i	0.110932i
$479$	− 631.947i	− 1.31930i	−0.751571	−	0.659652i	$-0.770704\pi$
$479$	− 631.947i	− 1.31930i	0.751571	−	0.659652i	$-0.229296\pi$
$480$	0	0
$481$	−9.63104	−0.0200230
$482$	216.910	0.450022
$483$	0	0
$484$	231.166	0.477616
$485$	0	0
$486$	0	0
$487$	− 241.105i	− 0.495081i	−0.968877	−	0.247541i	$-0.920378\pi$
$487$	− 241.105i	− 0.495081i	0.968877	−	0.247541i	$-0.0796224\pi$
$488$	230.630	0.472602
$489$	0	0
$490$	0	0
$491$	176.708i	0.359893i	0.983676	+	0.179947i	$0.0575925\pi$
$491$	176.708i	0.359893i	−0.983676	+	0.179947i	$0.942407\pi$
$492$	0	0
$493$	15.7507i	0.0319487i
$494$	− 167.769i	− 0.339614i
$495$	0	0
$496$	−61.0821	−0.123149
$497$	−162.253	−0.326464
$498$	0	0
$499$	−431.360	−0.864449	−0.432224	−	0.901766i	$-0.642271\pi$
$499$	−431.360	−0.864449	−0.432224	+	0.901766i	$0.642271\pi$
$500$	0	0
$501$	0	0
$502$	− 171.848i	− 0.342326i
$503$	−138.491	−0.275330	−0.137665	−	0.990479i	$-0.543960\pi$
$503$	−138.491	−0.275330	−0.137665	+	0.990479i	$0.543960\pi$
$504$	0	0
$505$	0	0
$506$	− 114.237i	− 0.225764i
$507$	0	0
$508$	− 209.605i	− 0.412608i
$509$	− 693.349i	− 1.36218i	−0.732201	−	0.681089i	$-0.761507\pi$
$509$	− 693.349i	− 1.36218i	0.732201	−	0.681089i	$-0.238493\pi$
$510$	0	0
$511$	−54.3457	−0.106352
$512$	−22.6274	−0.0441942
$513$	0	0
$514$	−547.895	−1.06594
$515$	0	0
$516$	0	0
$517$	− 39.9230i	− 0.0772206i
$518$	−3.37064	−0.00650703
$519$	0	0
$520$	0	0
$521$	− 74.9273i	− 0.143814i	−0.997411	−	0.0719072i	$-0.977091\pi$
$521$	− 74.9273i	− 0.143814i	0.997411	−	0.0719072i	$-0.0229086\pi$
$522$	0	0
$523$	62.8549i	0.120182i	0.998193	+	0.0600908i	$0.0191390\pi$
$523$	62.8549i	0.120182i	−0.998193	+	0.0600908i	$0.980861\pi$
$524$	337.851i	0.644754i
$525$	0	0
$526$	324.846	0.617578
$527$	41.3089	0.0783849
$528$	0	0
$529$	675.544	1.27702
$530$	0	0
$531$	0	0
$532$	− 58.7154i	− 0.110367i
$533$	338.547	0.635173
$534$	0	0
$535$	0	0
$536$	130.709i	0.243859i
$537$	0	0
$538$	575.190i	1.06913i
$539$	− 16.2921i	− 0.0302265i
$540$	0	0
$541$	510.104	0.942890	0.471445	−	0.881895i	$-0.343733\pi$
$541$	510.104	0.942890	0.471445	+	0.881895i	$0.343733\pi$
$542$	−197.940	−0.365203
$543$	0	0
$544$	15.3026	0.0281297
$545$	0	0
$546$	0	0
$547$	587.829i	1.07464i	0.843378	+	0.537320i	$0.180563\pi$
$547$	587.829i	1.07464i	−0.843378	+	0.537320i	$0.819437\pi$
$548$	221.111	0.403486
$549$	0	0
$550$	0	0
$551$	64.6077i	0.117255i
$552$	0	0
$553$	− 206.599i	− 0.373596i
$554$	673.846i	1.21633i
$555$	0	0
$556$	−446.906	−0.803788
$557$	−730.298	−1.31113	−0.655564	−	0.755140i	$-0.727569\pi$
$557$	−730.298	−1.31113	−0.655564	+	0.755140i	$0.727569\pi$
$558$	0	0
$559$	−139.789	−0.250070
$560$	0	0
$561$	0	0
$562$	414.318i	0.737220i
$563$	−36.0516	−0.0640349	−0.0320174	−	0.999487i	$-0.510193\pi$
$563$	−36.0516	−0.0640349	−0.0320174	+	0.999487i	$0.510193\pi$
$564$	0	0
$565$	0	0
$566$	− 222.173i	− 0.392532i
$567$	0	0
$568$	− 173.456i	− 0.305379i
$569$	903.457i	1.58780i	0.608050	+	0.793899i	$0.291952\pi$
$569$	903.457i	1.58780i	−0.608050	+	0.793899i	$0.708048\pi$
$570$	0	0
$571$	167.817	0.293899	0.146950	−	0.989144i	$-0.453054\pi$
$571$	167.817	0.293899	0.146950	+	0.989144i	$0.453054\pi$
$572$	49.7661	0.0870037
$573$	0	0
$574$	118.484	0.206417
$575$	0	0
$576$	0	0
$577$	− 405.888i	− 0.703445i	−0.936104	−	0.351722i	$-0.885596\pi$
$577$	− 405.888i	− 0.703445i	0.936104	−	0.351722i	$-0.114404\pi$
$578$	398.359	0.689202
$579$	0	0
$580$	0	0
$581$	− 75.9687i	− 0.130755i
$582$	0	0
$583$	158.700i	0.272213i
$584$	− 58.0980i	− 0.0994829i
$585$	0	0
$586$	42.9674	0.0733231
$587$	202.539	0.345041	0.172520	−	0.985006i	$-0.444809\pi$
$587$	202.539	0.345041	0.172520	+	0.985006i	$0.444809\pi$
$588$	0	0
$589$	169.444	0.287682
$590$	0	0
$591$	0	0
$592$	− 3.60337i	− 0.00608677i
$593$	−1080.22	−1.82162	−0.910812	−	0.412821i	$-0.864543\pi$
$593$	−1080.22	−1.82162	−0.910812	+	0.412821i	$0.864543\pi$
$594$	0	0
$595$	0	0
$596$	160.863i	0.269904i
$597$	0	0
$598$	524.748i	0.877505i
$599$	− 110.028i	− 0.183687i	−0.995773	−	0.0918434i	$-0.970724\pi$
$599$	− 110.028i	− 0.183687i	0.995773	−	0.0918434i	$-0.0292759\pi$
$600$	0	0
$601$	−667.302	−1.11032	−0.555160	−	0.831744i	$-0.687343\pi$
$601$	−667.302	−1.11032	−0.555160	+	0.831744i	$0.687343\pi$
$602$	−48.9229	−0.0812673
$603$	0	0
$604$	−276.515	−0.457807
$605$	0	0
$606$	0	0
$607$	354.193i	0.583514i	0.956492	+	0.291757i	$0.0942399\pi$
$607$	354.193i	0.583514i	−0.956492	+	0.291757i	$0.905760\pi$
$608$	62.7694	0.103239
$609$	0	0
$610$	0	0
$611$	183.387i	0.300143i
$612$	0	0
$613$	374.939i	0.611646i	0.952088	+	0.305823i	$0.0989315\pi$
$613$	374.939i	0.611646i	−0.952088	+	0.305823i	$0.901068\pi$
$614$	− 492.241i	− 0.801696i
$615$	0	0
$616$	17.4170	0.0282743
$617$	−157.220	−0.254814	−0.127407	−	0.991850i	$-0.540665\pi$
$617$	−157.220	−0.254814	−0.127407	+	0.991850i	$0.540665\pi$
$618$	0	0
$619$	−977.265	−1.57878	−0.789390	−	0.613892i	$-0.789603\pi$
$619$	−977.265	−1.57878	−0.789390	+	0.613892i	$0.789603\pi$
$620$	0	0
$621$	0	0
$622$	− 32.9613i	− 0.0529925i
$623$	362.164	0.581323
$624$	0	0
$625$	0	0
$626$	− 767.999i	− 1.22684i
$627$	0	0
$628$	3.01064i	0.00479401i
$629$	2.43690i	0.00387424i
$630$	0	0
$631$	624.635	0.989912	0.494956	−	0.868918i	$-0.335184\pi$
$631$	624.635	0.989912	0.494956	+	0.868918i	$0.335184\pi$
$632$	220.863	0.349467
$633$	0	0
$634$	−614.262	−0.968867
$635$	0	0
$636$	0	0
$637$	74.8381i	0.117485i
$638$	−19.1648	−0.0300389
$639$	0	0
$640$	0	0
$641$	− 535.603i	− 0.835574i	−0.908545	−	0.417787i	$-0.862806\pi$
$641$	− 535.603i	− 0.835574i	0.908545	−	0.417787i	$-0.137194\pi$
$642$	0	0
$643$	− 974.527i	− 1.51559i	−0.652490	−	0.757797i	$-0.726276\pi$
$643$	− 974.527i	− 1.51559i	0.652490	−	0.757797i	$-0.273724\pi$
$644$	183.650i	0.285170i
$645$	0	0
$646$	−42.4500	−0.0657120
$647$	−132.916	−0.205435	−0.102717	−	0.994711i	$-0.532754\pi$
$647$	−132.916	−0.205435	−0.102717	+	0.994711i	$0.532754\pi$
$648$	0	0
$649$	79.9971	0.123262
$650$	0	0
$651$	0	0
$652$	− 445.964i	− 0.683993i
$653$	45.5996	0.0698310	0.0349155	−	0.999390i	$-0.488884\pi$
$653$	45.5996	0.0698310	0.0349155	+	0.999390i	$0.488884\pi$
$654$	0	0
$655$	0	0
$656$	126.664i	0.193086i
$657$	0	0
$658$	64.1813i	0.0975399i
$659$	124.398i	0.188769i	0.995536	+	0.0943843i	$0.0300882\pi$
$659$	124.398i	0.188769i	−0.995536	+	0.0943843i	$0.969912\pi$
$660$	0	0
$661$	514.066	0.777710	0.388855	−	0.921299i	$-0.372871\pi$
$661$	514.066	0.777710	0.388855	+	0.921299i	$0.372871\pi$
$662$	−125.090	−0.188957
$663$	0	0
$664$	81.2139	0.122310
$665$	0	0
$666$	0	0
$667$	− 202.079i	− 0.302968i
$668$	240.966	0.360728
$669$	0	0
$670$	0	0
$671$	− 189.780i	− 0.282831i
$672$	0	0
$673$	− 441.681i	− 0.656287i	−0.944628	−	0.328144i	$-0.893577\pi$
$673$	− 441.681i	− 0.656287i	0.944628	−	0.328144i	$-0.106423\pi$
$674$	595.962i	0.884216i
$675$	0	0
$676$	109.398	0.161832
$677$	−103.462	−0.152825	−0.0764124	−	0.997076i	$-0.524347\pi$
$677$	−103.462	−0.152825	−0.0764124	+	0.997076i	$0.524347\pi$
$678$	0	0
$679$	−26.5091	−0.0390414
$680$	0	0
$681$	0	0
$682$	50.2630i	0.0736994i
$683$	−576.481	−0.844042	−0.422021	−	0.906586i	$-0.638679\pi$
$683$	−576.481	−0.844042	−0.422021	+	0.906586i	$0.638679\pi$
$684$	0	0
$685$	0	0
$686$	26.1916i	0.0381802i
$687$	0	0
$688$	− 52.3008i	− 0.0760186i
$689$	− 728.991i	− 1.05804i
$690$	0	0
$691$	−546.270	−0.790549	−0.395275	−	0.918563i	$-0.629351\pi$
$691$	−546.270	−0.790549	−0.395275	+	0.918563i	$0.629351\pi$
$692$	200.552	0.289816
$693$	0	0
$694$	122.276	0.176191
$695$	0	0
$696$	0	0
$697$	− 85.6611i	− 0.122900i
$698$	−25.2839	−0.0362233
$699$	0	0
$700$	0	0
$701$	− 248.599i	− 0.354634i	−0.984154	−	0.177317i	$-0.943258\pi$
$701$	− 248.599i	− 0.354634i	0.984154	−	0.177317i	$-0.0567419\pi$
$702$	0	0
$703$	9.99590i	0.0142189i
$704$	18.6196i	0.0264482i
$705$	0	0
$706$	−167.491	−0.237239
$707$	98.2246	0.138932
$708$	0	0
$709$	463.405	0.653604	0.326802	−	0.945093i	$-0.394029\pi$
$709$	463.405	0.653604	0.326802	+	0.945093i	$0.394029\pi$
$710$	0	0
$711$	0	0
$712$	387.170i	0.543778i
$713$	−529.987	−0.743320
$714$	0	0
$715$	0	0
$716$	− 182.913i	− 0.255465i
$717$	0	0
$718$	159.524i	0.222178i
$719$	5.86223i	0.00815330i	0.999992	+	0.00407665i	$0.00129764\pi$
$719$	5.86223i	0.00815330i	−0.999992	+	0.00407665i	$0.998702\pi$
$720$	0	0
$721$	283.879	0.393730
$722$	336.406	0.465936
$723$	0	0
$724$	260.795	0.360214
$725$	0	0
$726$	0	0
$727$	840.247i	1.15577i	0.816117	+	0.577887i	$0.196123\pi$
$727$	840.247i	1.15577i	−0.816117	+	0.577887i	$0.803877\pi$
$728$	−80.0053	−0.109897
$729$	0	0
$730$	0	0
$731$	35.3702i	0.0483860i
$732$	0	0
$733$	227.202i	0.309962i	0.987917	+	0.154981i	$0.0495316\pi$
$733$	227.202i	0.309962i	−0.987917	+	0.154981i	$0.950468\pi$
$734$	854.497i	1.16417i
$735$	0	0
$736$	−196.330	−0.266752
$737$	107.557	0.145939
$738$	0	0
$739$	649.346	0.878682	0.439341	−	0.898320i	$-0.355212\pi$
$739$	649.346	0.878682	0.439341	+	0.898320i	$0.355212\pi$
$740$	0	0
$741$	0	0
$742$	− 255.130i	− 0.343841i
$743$	681.019	0.916580	0.458290	−	0.888803i	$-0.348462\pi$
$743$	681.019	0.916580	0.458290	+	0.888803i	$0.348462\pi$
$744$	0	0
$745$	0	0
$746$	188.604i	0.252820i
$747$	0	0
$748$	− 12.5921i	− 0.0168344i
$749$	376.270i	0.502363i
$750$	0	0
$751$	507.471	0.675727	0.337863	−	0.941195i	$-0.390296\pi$
$751$	507.471	0.675727	0.337863	+	0.941195i	$0.390296\pi$
$752$	−68.6127	−0.0912402
$753$	0	0
$754$	88.0340	0.116756
$755$	0	0
$756$	0	0
$757$	1503.28i	1.98585i	0.118764	+	0.992923i	$0.462107\pi$
$757$	1503.28i	1.98585i	−0.118764	+	0.992923i	$0.537893\pi$
$758$	−393.932	−0.519699
$759$	0	0
$760$	0	0
$761$	− 71.4488i	− 0.0938880i	−0.998898	−	0.0469440i	$-0.985052\pi$
$761$	− 71.4488i	− 0.0938880i	0.998898	−	0.0469440i	$-0.0149482\pi$
$762$	0	0
$763$	125.839i	0.164927i
$764$	− 237.954i	− 0.311458i
$765$	0	0
$766$	−632.619	−0.825874
$767$	−367.468	−0.479098
$768$	0	0
$769$	1276.97	1.66055	0.830277	−	0.557352i	$-0.188182\pi$
$769$	1276.97	1.66055	0.830277	+	0.557352i	$0.188182\pi$
$770$	0	0
$771$	0	0
$772$	− 405.632i	− 0.525430i
$773$	−1105.18	−1.42973	−0.714866	−	0.699261i	$-0.753512\pi$
$773$	−1105.18	−1.42973	−0.714866	+	0.699261i	$0.753512\pi$
$774$	0	0
$775$	0	0
$776$	− 28.3395i	− 0.0365199i
$777$	0	0
$778$	− 73.2149i	− 0.0941065i
$779$	− 351.372i	− 0.451056i
$780$	0	0
$781$	−142.732	−0.182756
$782$	132.775	0.169789
$783$	0	0
$784$	−28.0000	−0.0357143
$785$	0	0
$786$	0	0
$787$	1158.39i	1.47191i	0.677029	+	0.735956i	$0.263267\pi$
$787$	1158.39i	1.47191i	−0.677029	+	0.735956i	$0.736733\pi$
$788$	591.735	0.750932
$789$	0	0
$790$	0	0
$791$	− 273.310i	− 0.345525i
$792$	0	0
$793$	871.757i	1.09931i
$794$	547.067i	0.689001i
$795$	0	0
$796$	−606.047	−0.761365
$797$	−387.467	−0.486157	−0.243078	−	0.970007i	$-0.578157\pi$
$797$	−387.467	−0.486157	−0.243078	+	0.970007i	$0.578157\pi$
$798$	0	0
$799$	46.4016	0.0580747
$800$	0	0
$801$	0	0
$802$	864.284i	1.07766i
$803$	−47.8074	−0.0595360
$804$	0	0
$805$	0	0
$806$	− 230.884i	− 0.286457i
$807$	0	0
$808$	105.007i	0.129959i
$809$	1535.11i	1.89754i	0.315964	+	0.948771i	$0.397672\pi$
$809$	1535.11i	1.89754i	−0.315964	+	0.948771i	$0.602328\pi$
$810$	0	0
$811$	564.588	0.696163	0.348082	−	0.937464i	$-0.386833\pi$
$811$	564.588	0.696163	0.348082	+	0.937464i	$0.386833\pi$
$812$	30.8099	0.0379432
$813$	0	0
$814$	−2.96512	−0.00364266
$815$	0	0
$816$	0	0
$817$	145.085i	0.177582i
$818$	−452.877	−0.553639
$819$	0	0
$820$	0	0
$821$	− 1060.19i	− 1.29134i	−0.763615	−	0.645672i	$-0.776577\pi$
$821$	− 1060.19i	− 1.29134i	0.763615	−	0.645672i	$-0.223423\pi$
$822$	0	0
$823$	− 933.052i	− 1.13372i	−0.823814	−	0.566860i	$-0.808158\pi$
$823$	− 933.052i	− 1.13372i	0.823814	−	0.566860i	$-0.191842\pi$
$824$	303.480i	0.368300i
$825$	0	0
$826$	−128.605	−0.155697
$827$	597.367	0.722330	0.361165	−	0.932502i	$-0.382379\pi$
$827$	597.367	0.722330	0.361165	+	0.932502i	$0.382379\pi$
$828$	0	0
$829$	−1178.19	−1.42122	−0.710609	−	0.703587i	$-0.751580\pi$
$829$	−1178.19	−1.42122	−0.710609	+	0.703587i	$0.751580\pi$
$830$	0	0
$831$	0	0
$832$	− 85.5292i	− 0.102800i
$833$	18.9360	0.0227322
$834$	0	0
$835$	0	0
$836$	− 51.6514i	− 0.0617840i
$837$	0	0
$838$	661.174i	0.788991i
$839$	693.857i	0.827005i	0.910503	+	0.413503i	$0.135695\pi$
$839$	693.857i	0.827005i	−0.910503	+	0.413503i	$0.864305\pi$
$840$	0	0
$841$	807.098	0.959689
$842$	−249.747	−0.296612
$843$	0	0
$844$	468.146	0.554676
$845$	0	0
$846$	0	0
$847$	305.804i	0.361044i
$848$	272.745	0.321634
$849$	0	0
$850$	0	0
$851$	− 31.2651i	− 0.0367393i
$852$	0	0
$853$	− 166.782i	− 0.195524i	−0.995210	−	0.0977619i	$-0.968832\pi$
$853$	− 166.782i	− 0.195524i	0.995210	−	0.0977619i	$-0.0311683\pi$
$854$	305.095i	0.357254i
$855$	0	0
$856$	−402.249	−0.469917
$857$	−658.616	−0.768514	−0.384257	−	0.923226i	$-0.625542\pi$
$857$	−658.616	−0.768514	−0.384257	+	0.923226i	$0.625542\pi$
$858$	0	0
$859$	1267.25	1.47527	0.737633	−	0.675202i	$-0.235943\pi$
$859$	1267.25	1.47527	0.737633	+	0.675202i	$0.235943\pi$
$860$	0	0
$861$	0	0
$862$	225.307i	0.261377i
$863$	−961.019	−1.11358	−0.556790	−	0.830654i	$-0.687967\pi$
$863$	−961.019	−1.11358	−0.556790	+	0.830654i	$0.687967\pi$
$864$	0	0
$865$	0	0
$866$	439.596i	0.507617i
$867$	0	0
$868$	− 80.8041i	− 0.0930922i
$869$	− 181.743i	− 0.209141i
$870$	0	0
$871$	−494.065	−0.567239
$872$	−134.527	−0.154275
$873$	0	0
$874$	544.627	0.623144
$875$	0	0
$876$	0	0
$877$	− 33.2355i	− 0.0378968i	−0.999820	−	0.0189484i	$-0.993968\pi$
$877$	− 33.2355i	− 0.0378968i	0.999820	−	0.0189484i	$-0.00603182\pi$
$878$	−1135.34	−1.29310
$879$	0	0
$880$	0	0
$881$	88.5981i	0.100565i	0.998735	+	0.0502827i	$0.0160122\pi$
$881$	88.5981i	0.100565i	−0.998735	+	0.0502827i	$0.983988\pi$
$882$	0	0
$883$	477.823i	0.541135i	0.962701	+	0.270568i	$0.0872114\pi$
$883$	477.823i	0.541135i	−0.962701	+	0.270568i	$0.912789\pi$
$884$	57.8421i	0.0654322i
$885$	0	0
$886$	928.122	1.04754
$887$	1317.14	1.48493	0.742467	−	0.669882i	$-0.233655\pi$
$887$	1317.14	1.48493	0.742467	+	0.669882i	$0.233655\pi$
$888$	0	0
$889$	277.281	0.311902
$890$	0	0
$891$	0	0
$892$	− 602.052i	− 0.674946i
$893$	190.334	0.213141
$894$	0	0
$895$	0	0
$896$	− 29.9333i	− 0.0334077i
$897$	0	0
$898$	518.786i	0.577713i
$899$	88.9130i	0.0989021i
$900$	0	0
$901$	−184.453	−0.204721
$902$	104.229	0.115553
$903$	0	0
$904$	292.181	0.323209
$905$	0	0
$906$	0	0
$907$	201.813i	0.222506i	0.993792	+	0.111253i	$0.0354863\pi$
$907$	201.813i	0.222506i	−0.993792	+	0.111253i	$0.964514\pi$
$908$	−195.556	−0.215371
$909$	0	0
$910$	0	0
$911$	1188.08i	1.30415i	0.758153	+	0.652076i	$0.226102\pi$
$911$	1188.08i	1.30415i	−0.758153	+	0.652076i	$0.773898\pi$
$912$	0	0
$913$	− 66.8290i	− 0.0731971i
$914$	901.789i	0.986640i
$915$	0	0
$916$	−143.494	−0.156652
$917$	−446.935	−0.487388
$918$	0	0
$919$	−982.856	−1.06948	−0.534742	−	0.845015i	$-0.679591\pi$
$919$	−982.856	−1.06948	−0.534742	+	0.845015i	$0.679591\pi$
$920$	0	0
$921$	0	0
$922$	836.941i	0.907745i
$923$	655.644	0.710340
$924$	0	0
$925$	0	0
$926$	1272.65i	1.37435i
$927$	0	0
$928$	32.9371i	0.0354926i
$929$	− 630.846i	− 0.679060i	−0.940595	−	0.339530i	$-0.889732\pi$
$929$	− 630.846i	− 0.679060i	0.940595	−	0.339530i	$-0.110268\pi$
$930$	0	0
$931$	77.6732	0.0834299
$932$	−642.792	−0.689691
$933$	0	0
$934$	−592.131	−0.633974
$935$	0	0
$936$	0	0
$937$	1036.23i	1.10591i	0.833212	+	0.552953i	$0.186499\pi$
$937$	1036.23i	1.10591i	−0.833212	+	0.552953i	$0.813501\pi$
$938$	−172.911	−0.184340
$939$	0	0
$940$	0	0
$941$	326.428i	0.346894i	0.984843	+	0.173447i	$0.0554906\pi$
$941$	326.428i	0.346894i	−0.984843	+	0.173447i	$0.944509\pi$
$942$	0	0
$943$	1099.02i	1.16545i
$944$	− 137.485i	− 0.145641i
$945$	0	0
$946$	−43.0371	−0.0454937
$947$	−1020.01	−1.07709	−0.538547	−	0.842595i	$-0.681027\pi$
$947$	−1020.01	−1.07709	−0.538547	+	0.842595i	$0.681027\pi$
$948$	0	0
$949$	219.604	0.231406
$950$	0	0
$951$	0	0
$952$	20.2434i	0.0212641i
$953$	−268.365	−0.281600	−0.140800	−	0.990038i	$-0.544968\pi$
$953$	−268.365	−0.281600	−0.140800	+	0.990038i	$0.544968\pi$
$954$	0	0
$955$	0	0
$956$	− 74.9897i	− 0.0784411i
$957$	0	0
$958$	893.708i	0.932889i
$959$	292.502i	0.305007i
$960$	0	0
$961$	−727.811	−0.757347
$962$	13.6203	0.0141584
$963$	0	0
$964$	−306.758	−0.318213
$965$	0	0
$966$	0	0
$967$	963.369i	0.996245i	0.867107	+	0.498123i	$0.165977\pi$
$967$	963.369i	0.996245i	−0.867107	+	0.498123i	$0.834023\pi$
$968$	−326.918	−0.337725
$969$	0	0
$970$	0	0
$971$	− 321.620i	− 0.331225i	−0.986191	−	0.165613i	$-0.947040\pi$
$971$	− 321.620i	− 0.331225i	0.986191	−	0.165613i	$-0.0529601\pi$
$972$	0	0
$973$	− 591.201i	− 0.607607i
$974$	340.974i	0.350075i
$975$	0	0
$976$	−326.160	−0.334180
$977$	−1124.21	−1.15068	−0.575339	−	0.817915i	$-0.695130\pi$
$977$	−1124.21	−1.15068	−0.575339	+	0.817915i	$0.695130\pi$
$978$	0	0
$979$	318.593	0.325427
$980$	0	0
$981$	0	0
$982$	− 249.902i	− 0.254483i
$983$	−844.372	−0.858974	−0.429487	−	0.903073i	$-0.641306\pi$
$983$	−844.372	−0.858974	−0.429487	+	0.903073i	$0.641306\pi$
$984$	0	0
$985$	0	0
$986$	− 22.2749i	− 0.0225911i
$987$	0	0
$988$	237.262i	0.240144i
$989$	− 453.795i	− 0.458842i
$990$	0	0
$991$	−1038.57	−1.04801	−0.524003	−	0.851717i	$-0.675562\pi$
$991$	−1038.57	−1.04801	−0.524003	+	0.851717i	$0.675562\pi$
$992$	86.3832	0.0870798
$993$	0	0
$994$	229.460	0.230845
$995$	0	0
$996$	0	0
$997$	1050.74i	1.05390i	0.849896	+	0.526950i	$0.176664\pi$
$997$	1050.74i	1.05390i	−0.849896	+	0.526950i	$0.823336\pi$
$998$	610.035	0.611258
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.3.c.d.449.5		16
3.2	odd	2	inner	3150.3.c.d.449.15		16
5.2	odd	4		3150.3.e.i.701.1		8
5.3	odd	4		630.3.e.a.71.8	yes	8
5.4	even	2	inner	3150.3.c.d.449.10		16
15.2	even	4		3150.3.e.i.701.5		8
15.8	even	4		630.3.e.a.71.2	✓	8
15.14	odd	2	inner	3150.3.c.d.449.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
630.3.e.a.71.2	✓	8	15.8	even	4
630.3.e.a.71.8	yes	8	5.3	odd	4
3150.3.c.d.449.4		16	15.14	odd	2	inner
3150.3.c.d.449.5		16	1.1	even	1	trivial
3150.3.c.d.449.10		16	5.4	even	2	inner
3150.3.c.d.449.15		16	3.2	odd	2	inner
3150.3.e.i.701.1		8	5.2	odd	4
3150.3.e.i.701.5		8	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3150.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +2.64575i q^{7} -2.82843 q^{8} +2.32744i q^{11} -10.6912i q^{13} -3.74166i q^{14} +4.00000 q^{16} -2.70514 q^{17} -11.0962 q^{19} -3.29150i q^{22} +34.7065 q^{23} +15.1196i q^{26} +5.29150i q^{28} -5.82252i q^{29} -15.2705 q^{31} -5.65685 q^{32} +3.82564 q^{34} -0.900842i q^{37} +15.6924 q^{38} +31.6661i q^{41} -13.0752i q^{43} +4.65489i q^{44} -49.0825 q^{46} -17.1532 q^{47} -7.00000 q^{49} -21.3823i q^{52} +68.1864 q^{53} -7.48331i q^{56} +8.23429i q^{58} -34.3712i q^{59} -81.5400 q^{61} +21.5958 q^{62} +8.00000 q^{64} -46.2125i q^{67} -5.41027 q^{68} +61.3258i q^{71} +20.5408i q^{73} +1.27398i q^{74} -22.1923 q^{76} -6.15784 q^{77} -78.0870 q^{79} -44.7826i q^{82} -28.7135 q^{83} +18.4911i q^{86} -6.58301i q^{88} -136.885i q^{89} +28.2861 q^{91} +69.4131 q^{92} +24.2582 q^{94} +10.0195i q^{97} +9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} + 64 q^{16} - 160 q^{19} - 224 q^{31} + 64 q^{34} - 64 q^{46} - 112 q^{49} - 160 q^{61} + 128 q^{64} - 320 q^{76} + 128 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 + 64 * q^16 - 160 * q^19 - 224 * q^31 + 64 * q^34 - 64 * q^46 - 112 * q^49 - 160 * q^61 + 128 * q^64 - 320 * q^76 + 128 * q^94

Introduction

L-functions

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Varieties

Fields

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Database

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