LMFDB - Embedded newform 450.7.b.a.449.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [450,7,Mod(449,450)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("450.449"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(450, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	450.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,128,0,0,0,0,0,0,0,0,0,0,0,4096,0,0,-22976] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$103.524337629$
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:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	450.449
Dual form			450.7.b.a.449.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.65685 q^{2} +32.0000 q^{4} -484.000i q^{7} -181.019 q^{8} +1340.67i q^{11} -3368.00i q^{13} +2737.92i q^{14} +1024.00 q^{16} -12.7279 q^{17} -5744.00 q^{19} -7584.00i q^{22} -3377.14 q^{23} +19052.3i q^{26} -15488.0i q^{28} -29354.8i q^{29} -39796.0 q^{31} -5792.62 q^{32} +72.0000 q^{34} +52526.0i q^{37} +32493.0 q^{38} -37042.5i q^{41} -3800.00i q^{43} +42901.6i q^{44} +19104.0 q^{46} -76791.8 q^{47} -116607. q^{49} -107776. i q^{52} +238738. q^{53} +87613.4i q^{56} +166056. i q^{58} +249841. i q^{59} +13250.0 q^{61} +225120. q^{62} +32768.0 q^{64} +168968. i q^{67} -407.294 q^{68} -531467. i q^{71} -236144. i q^{73} -297132. i q^{74} -183808. q^{76} +648886. q^{77} +35116.0 q^{79} +209544. i q^{82} -10980.0 q^{83} +21496.0i q^{86} -242688. i q^{88} +129328. i q^{89} -1.63011e6 q^{91} -108069. q^{92} +434400. q^{94} -321424. i q^{97} +659629. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 128 q^{4} + 4096 q^{16} - 22976 q^{19} - 159184 q^{31} + 288 q^{34} + 76416 q^{46} - 466428 q^{49} + 53000 q^{61} + 131072 q^{64} - 735232 q^{76} + 140464 q^{79} - 6520448 q^{91} + 1737600 q^{94}+O(q^{100}) $ 4 * q + 128 * q^4 + 4096 * q^16 - 22976 * q^19 - 159184 * q^31 + 288 * q^34 + 76416 * q^46 - 466428 * q^49 + 53000 * q^61 + 131072 * q^64 - 735232 * q^76 + 140464 * q^79 - 6520448 * q^91 + 1737600 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−5.65685	−0.707107
$2$	−5.65685	−0.707107
$3$	0	0
$3$	0	0
$4$	32.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 484.000i	− 1.41108i	−0.708671	−	0.705539i	$-0.750705\pi$
$7$	− 484.000i	− 1.41108i	0.708671	−	0.705539i	$-0.249295\pi$
$8$	−181.019	−0.353553
$9$	0	0
$10$	0	0
$11$	1340.67i	1.00727i	0.863917	+	0.503634i	$0.168004\pi$
$11$	1340.67i	1.00727i	−0.863917	+	0.503634i	$0.831996\pi$
$12$	0	0
$13$	− 3368.00i	− 1.53300i	−0.642245	−	0.766500i	$-0.721997\pi$
$13$	− 3368.00i	− 1.53300i	0.642245	−	0.766500i	$-0.278003\pi$
$14$	2737.92i	0.997783i
$15$	0	0
$16$	1024.00	0.250000
$17$	−12.7279	−0.00259066	−0.00129533	−	0.999999i	$-0.500412\pi$
$17$	−12.7279	−0.00259066	−0.00129533	+	0.999999i	$0.500412\pi$
$18$	0	0
$19$	−5744.00	−0.837440	−0.418720	−	0.908115i	$-0.637521\pi$
$19$	−5744.00	−0.837440	−0.418720	+	0.908115i	$0.637521\pi$
$20$	0	0
$21$	0	0
$22$	− 7584.00i	− 0.712246i
$23$	−3377.14	−0.277566	−0.138783	−	0.990323i	$-0.544319\pi$
$23$	−3377.14	−0.277566	−0.138783	+	0.990323i	$0.544319\pi$
$24$	0	0
$25$	0	0
$26$	19052.3i	1.08399i
$27$	0	0
$28$	− 15488.0i	− 0.705539i
$29$	− 29354.8i	− 1.20361i	−0.798643	−	0.601805i	$-0.794449\pi$
$29$	− 29354.8i	− 1.20361i	0.798643	−	0.601805i	$-0.205551\pi$
$30$	0	0
$31$	−39796.0	−1.33584	−0.667920	−	0.744233i	$-0.732815\pi$
$31$	−39796.0	−1.33584	−0.667920	+	0.744233i	$0.732815\pi$
$32$	−5792.62	−0.176777
$33$	0	0
$34$	72.0000	0.00183187
$35$	0	0
$36$	0	0
$37$	52526.0i	1.03698i	0.855085	+	0.518489i	$0.173505\pi$
$37$	52526.0i	1.03698i	−0.855085	+	0.518489i	$0.826495\pi$
$38$	32493.0	0.592159
$39$	0	0
$40$	0	0
$41$	− 37042.5i	− 0.537463i	−0.963215	−	0.268732i	$-0.913396\pi$
$41$	− 37042.5i	− 0.537463i	0.963215	−	0.268732i	$-0.0866045\pi$
$42$	0	0
$43$	− 3800.00i	− 0.0477945i	−0.999714	−	0.0238973i	$-0.992393\pi$
$43$	− 3800.00i	− 0.0477945i	0.999714	−	0.0238973i	$-0.00760746\pi$
$44$	42901.6i	0.503634i
$45$	0	0
$46$	19104.0	0.196269
$47$	−76791.8	−0.739641	−0.369821	−	0.929103i	$-0.620581\pi$
$47$	−76791.8	−0.739641	−0.369821	+	0.929103i	$0.620581\pi$
$48$	0	0
$49$	−116607.	−0.991143
$50$	0	0
$51$	0	0
$52$	− 107776.i	− 0.766500i
$53$	238738.	1.60359	0.801795	−	0.597599i	$-0.203879\pi$
$53$	238738.	1.60359	0.801795	+	0.597599i	$0.203879\pi$
$54$	0	0
$55$	0	0
$56$	87613.4i	0.498892i
$57$	0	0
$58$	166056.i	0.851080i
$59$	249841.i	1.21649i	0.793751	+	0.608243i	$0.208125\pi$
$59$	249841.i	1.21649i	−0.793751	+	0.608243i	$0.791875\pi$
$60$	0	0
$61$	13250.0	0.0583749	0.0291875	−	0.999574i	$-0.490708\pi$
$61$	13250.0	0.0583749	0.0291875	+	0.999574i	$0.490708\pi$
$62$	225120.	0.944581
$63$	0	0
$64$	32768.0	0.125000
$65$	0	0
$66$	0	0
$67$	168968.i	0.561798i	0.959737	+	0.280899i	$0.0906326\pi$
$67$	168968.i	0.561798i	−0.959737	+	0.280899i	$0.909367\pi$
$68$	−407.294	−0.00129533
$69$	0	0
$70$	0	0
$71$	− 531467.i	− 1.48491i	−0.669894	−	0.742457i	$-0.733660\pi$
$71$	− 531467.i	− 1.48491i	0.669894	−	0.742457i	$-0.266340\pi$
$72$	0	0
$73$	− 236144.i	− 0.607027i	−0.952827	−	0.303514i	$-0.901840\pi$
$73$	− 236144.i	− 0.607027i	0.952827	−	0.303514i	$-0.0981598\pi$
$74$	− 297132.i	− 0.733254i
$75$	0	0
$76$	−183808.	−0.418720
$77$	648886.	1.42134
$78$	0	0
$79$	35116.0	0.0712236	0.0356118	−	0.999366i	$-0.488662\pi$
$79$	35116.0	0.0712236	0.0356118	+	0.999366i	$0.488662\pi$
$80$	0	0
$81$	0	0
$82$	209544.i	0.380044i
$83$	−10980.0	−0.0192029	−0.00960144	−	0.999954i	$-0.503056\pi$
$83$	−10980.0	−0.0192029	−0.00960144	+	0.999954i	$0.503056\pi$
$84$	0	0
$85$	0	0
$86$	21496.0i	0.0337958i
$87$	0	0
$88$	− 242688.i	− 0.356123i
$89$	129328.i	0.183453i	0.995784	+	0.0917263i	$0.0292385\pi$
$89$	129328.i	0.183453i	−0.995784	+	0.0917263i	$0.970762\pi$
$90$	0	0
$91$	−1.63011e6	−2.16318
$92$	−108069.	−0.138783
$93$	0	0
$94$	434400.	0.523005
$95$	0	0
$96$	0	0
$97$	− 321424.i	− 0.352179i	−0.984374	−	0.176089i	$-0.943655\pi$
$97$	− 321424.i	− 0.352179i	0.984374	−	0.176089i	$-0.0563448\pi$
$98$	659629.	0.700844
$99$	0	0
$100$	0	0
$101$	668780.i	0.649111i	0.945867	+	0.324556i	$0.105215\pi$
$101$	668780.i	0.649111i	−0.945867	+	0.324556i	$0.894785\pi$
$102$	0	0
$103$	− 1.99341e6i	− 1.82425i	−0.409907	−	0.912127i	$-0.634439\pi$
$103$	− 1.99341e6i	− 1.82425i	0.409907	−	0.912127i	$-0.365561\pi$
$104$	609673.i	0.541997i
$105$	0	0
$106$	−1.35050e6	−1.13391
$107$	260668.	0.212783	0.106391	−	0.994324i	$-0.466070\pi$
$107$	260668.	0.212783	0.106391	+	0.994324i	$0.466070\pi$
$108$	0	0
$109$	−194456.	−0.150156	−0.0750779	−	0.997178i	$-0.523921\pi$
$109$	−194456.	−0.150156	−0.0750779	+	0.997178i	$0.523921\pi$
$110$	0	0
$111$	0	0
$112$	− 495616.i	− 0.352770i
$113$	−821897.	−0.569616	−0.284808	−	0.958585i	$-0.591930\pi$
$113$	−821897.	−0.569616	−0.284808	+	0.958585i	$0.591930\pi$
$114$	0	0
$115$	0	0
$116$	− 939355.i	− 0.601805i
$117$	0	0
$118$	− 1.41331e6i	− 0.860185i
$119$	6160.31i	0.00365563i
$120$	0	0
$121$	−25847.0	−0.0145900
$122$	−74953.3	−0.0412773
$123$	0	0
$124$	−1.27347e6	−0.667920
$125$	0	0
$126$	0	0
$127$	3.05721e6i	1.49250i	0.665666	+	0.746250i	$0.268148\pi$
$127$	3.05721e6i	1.49250i	−0.665666	+	0.746250i	$0.731852\pi$
$128$	−185364.	−0.0883883
$129$	0	0
$130$	0	0
$131$	3.07388e6i	1.36733i	0.729797	+	0.683664i	$0.239615\pi$
$131$	3.07388e6i	1.36733i	−0.729797	+	0.683664i	$0.760385\pi$
$132$	0	0
$133$	2.78010e6i	1.18169i
$134$	− 955827.i	− 0.397251i
$135$	0	0
$136$	2304.00	0.000915937 0
$137$	4.48412e6	1.74388	0.871938	−	0.489617i	$-0.162863\pi$
$137$	4.48412e6	1.74388	0.871938	+	0.489617i	$0.162863\pi$
$138$	0	0
$139$	1.09233e6	0.406732	0.203366	−	0.979103i	$-0.434812\pi$
$139$	1.09233e6	0.406732	0.203366	+	0.979103i	$0.434812\pi$
$140$	0	0
$141$	0	0
$142$	3.00643e6i	1.04999i
$143$	4.51539e6	1.54414
$144$	0	0
$145$	0	0
$146$	1.33583e6i	0.429233i
$147$	0	0
$148$	1.68083e6i	0.518489i
$149$	2.22087e6i	0.671375i	0.941973	+	0.335687i	$0.108969\pi$
$149$	2.22087e6i	0.671375i	−0.941973	+	0.335687i	$0.891031\pi$
$150$	0	0
$151$	−4.07871e6	−1.18465	−0.592327	−	0.805697i	$-0.701791\pi$
$151$	−4.07871e6	−1.18465	−0.592327	+	0.805697i	$0.701791\pi$
$152$	1.03978e6	0.296080
$153$	0	0
$154$	−3.67066e6	−1.00504
$155$	0	0
$156$	0	0
$157$	6.15568e6i	1.59066i	0.606178	+	0.795329i	$0.292702\pi$
$157$	6.15568e6i	1.59066i	−0.606178	+	0.795329i	$0.707298\pi$
$158$	−198646.	−0.0503627
$159$	0	0
$160$	0	0
$161$	1.63454e6i	0.391667i
$162$	0	0
$163$	− 800696.i	− 0.184886i	−0.995718	−	0.0924432i	$-0.970532\pi$
$163$	− 800696.i	− 0.184886i	0.995718	−	0.0924432i	$-0.0294676\pi$
$164$	− 1.18536e6i	− 0.268732i
$165$	0	0
$166$	62112.0	0.0135785
$167$	−4.80467e6	−1.03161	−0.515804	−	0.856707i	$-0.672507\pi$
$167$	−4.80467e6	−1.03161	−0.515804	+	0.856707i	$0.672507\pi$
$168$	0	0
$169$	−6.51661e6	−1.35009
$170$	0	0
$171$	0	0
$172$	− 121600.i	− 0.0238973i
$173$	−3.56992e6	−0.689478	−0.344739	−	0.938699i	$-0.612033\pi$
$173$	−3.56992e6	−0.689478	−0.344739	+	0.938699i	$0.612033\pi$
$174$	0	0
$175$	0	0
$176$	1.37285e6i	0.251817i
$177$	0	0
$178$	− 731592.i	− 0.129721i
$179$	7.43698e6i	1.29669i	0.761345	+	0.648347i	$0.224539\pi$
$179$	7.43698e6i	1.29669i	−0.761345	+	0.648347i	$0.775461\pi$
$180$	0	0
$181$	−1.03812e7	−1.75070	−0.875350	−	0.483491i	$-0.839369\pi$
$181$	−1.03812e7	−1.75070	−0.875350	+	0.483491i	$0.839369\pi$
$182$	9.22131e6	1.52960
$183$	0	0
$184$	611328.	0.0981343
$185$	0	0
$186$	0	0
$187$	− 17064.0i	− 0.00260949i
$188$	−2.45734e6	−0.369821
$189$	0	0
$190$	0	0
$191$	1.29941e7i	1.86485i	0.361360	+	0.932426i	$0.382313\pi$
$191$	1.29941e7i	1.86485i	−0.361360	+	0.932426i	$0.617687\pi$
$192$	0	0
$193$	3.93195e6i	0.546936i	0.961881	+	0.273468i	$0.0881708\pi$
$193$	3.93195e6i	0.546936i	−0.961881	+	0.273468i	$0.911829\pi$
$194$	1.81825e6i	0.249028i
$195$	0	0
$196$	−3.73142e6	−0.495572
$197$	−5.37967e6	−0.703651	−0.351825	−	0.936066i	$-0.614439\pi$
$197$	−5.37967e6	−0.703651	−0.351825	+	0.936066i	$0.614439\pi$
$198$	0	0
$199$	565900.	0.0718093	0.0359046	−	0.999355i	$-0.488569\pi$
$199$	565900.	0.0718093	0.0359046	+	0.999355i	$0.488569\pi$
$200$	0	0
$201$	0	0
$202$	− 3.78319e6i	− 0.458991i
$203$	−1.42077e7	−1.69839
$204$	0	0
$205$	0	0
$206$	1.12764e7i	1.28994i
$207$	0	0
$208$	− 3.44883e6i	− 0.383250i
$209$	− 7.70083e6i	− 0.843527i
$210$	0	0
$211$	−1.35165e7	−1.43885	−0.719427	−	0.694568i	$-0.755596\pi$
$211$	−1.35165e7	−1.43885	−0.719427	+	0.694568i	$0.755596\pi$
$212$	7.63960e6	0.801795
$213$	0	0
$214$	−1.47456e6	−0.150460
$215$	0	0
$216$	0	0
$217$	1.92613e7i	1.88497i
$218$	1.10001e6	0.106176
$219$	0	0
$220$	0	0
$221$	42867.6i	0.00397148i
$222$	0	0
$223$	5.35484e6i	0.482872i	0.970417	+	0.241436i	$0.0776183\pi$
$223$	5.35484e6i	0.482872i	−0.970417	+	0.241436i	$0.922382\pi$
$224$	2.80363e6i	0.249446i
$225$	0	0
$226$	4.64935e6	0.402779
$227$	1.36063e7	1.16322	0.581612	−	0.813466i	$-0.302422\pi$
$227$	1.36063e7	1.16322	0.581612	+	0.813466i	$0.302422\pi$
$228$	0	0
$229$	−4.34641e6	−0.361930	−0.180965	−	0.983490i	$-0.557922\pi$
$229$	−4.34641e6	−0.361930	−0.180965	+	0.983490i	$0.557922\pi$
$230$	0	0
$231$	0	0
$232$	5.31379e6i	0.425540i
$233$	−2.02333e7	−1.59956	−0.799778	−	0.600297i	$-0.795049\pi$
$233$	−2.02333e7	−1.59956	−0.799778	+	0.600297i	$0.795049\pi$
$234$	0	0
$235$	0	0
$236$	7.99490e6i	0.608243i
$237$	0	0
$238$	− 34848.0i	− 0.00258492i
$239$	2.03947e7i	1.49391i	0.664877	+	0.746953i	$0.268484\pi$
$239$	2.03947e7i	1.49391i	−0.664877	+	0.746953i	$0.731516\pi$
$240$	0	0
$241$	−3.12093e6	−0.222963	−0.111481	−	0.993767i	$-0.535560\pi$
$241$	−3.12093e6	−0.222963	−0.111481	+	0.993767i	$0.535560\pi$
$242$	146213.	0.0103167
$243$	0	0
$244$	424000.	0.0291875
$245$	0	0
$246$	0	0
$247$	1.93458e7i	1.28379i
$248$	7.20385e6	0.472291
$249$	0	0
$250$	0	0
$251$	5.09519e6i	0.322210i	0.986937	+	0.161105i	$0.0515058\pi$
$251$	5.09519e6i	0.322210i	−0.986937	+	0.161105i	$0.948494\pi$
$252$	0	0
$253$	− 4.52765e6i	− 0.279583i
$254$	− 1.72942e7i	− 1.05536i
$255$	0	0
$256$	1.04858e6	0.0625000
$257$	−1.44374e7	−0.850529	−0.425264	−	0.905069i	$-0.639819\pi$
$257$	−1.44374e7	−0.850529	−0.425264	+	0.905069i	$0.639819\pi$
$258$	0	0
$259$	2.54226e7	1.46326
$260$	0	0
$261$	0	0
$262$	− 1.73885e7i	− 0.966847i
$263$	3.12567e7	1.71821	0.859104	−	0.511801i	$-0.171022\pi$
$263$	3.12567e7	1.71821	0.859104	+	0.511801i	$0.171022\pi$
$264$	0	0
$265$	0	0
$266$	− 1.57266e7i	− 0.835584i
$267$	0	0
$268$	5.40698e6i	0.280899i
$269$	− 251338.i	− 0.0129122i	−0.999979	−	0.00645612i	$-0.997945\pi$
$269$	− 251338.i	− 0.0129122i	0.999979	−	0.00645612i	$-0.00205506\pi$
$270$	0	0
$271$	2.96399e7	1.48925	0.744627	−	0.667481i	$-0.232627\pi$
$271$	2.96399e7	1.48925	0.744627	+	0.667481i	$0.232627\pi$
$272$	−13033.4	−0.000647665 0
$273$	0	0
$274$	−2.53660e7	−1.23311
$275$	0	0
$276$	0	0
$277$	1.32213e7i	0.622062i	0.950400	+	0.311031i	$0.100674\pi$
$277$	1.32213e7i	0.622062i	−0.950400	+	0.311031i	$0.899326\pi$
$278$	−6.17914e6	−0.287603
$279$	0	0
$280$	0	0
$281$	− 6.12360e6i	− 0.275987i	−0.990433	−	0.137993i	$-0.955935\pi$
$281$	− 6.12360e6i	− 0.275987i	0.990433	−	0.137993i	$-0.0440652\pi$
$282$	0	0
$283$	6.74325e6i	0.297516i	0.988874	+	0.148758i	$0.0475275\pi$
$283$	6.74325e6i	0.297516i	−0.988874	+	0.148758i	$0.952473\pi$
$284$	− 1.70069e7i	− 0.742457i
$285$	0	0
$286$	−2.55429e7	−1.09187
$287$	−1.79286e7	−0.758403
$288$	0	0
$289$	−2.41374e7	−0.999993
$290$	0	0
$291$	0	0
$292$	− 7.55661e6i	− 0.303514i
$293$	−1.00239e7	−0.398505	−0.199253	−	0.979948i	$-0.563851\pi$
$293$	−1.00239e7	−0.398505	−0.199253	+	0.979948i	$0.563851\pi$
$294$	0	0
$295$	0	0
$296$	− 9.50822e6i	− 0.366627i
$297$	0	0
$298$	− 1.25632e7i	− 0.474734i
$299$	1.13742e7i	0.425508i
$300$	0	0
$301$	−1.83920e6	−0.0674418
$302$	2.30727e7	0.837677
$303$	0	0
$304$	−5.88186e6	−0.209360
$305$	0	0
$306$	0	0
$307$	− 5.23060e6i	− 0.180774i	−0.995907	−	0.0903871i	$-0.971190\pi$
$307$	− 5.23060e6i	− 0.180774i	0.995907	−	0.0903871i	$-0.0288104\pi$
$308$	2.07644e7	0.710668
$309$	0	0
$310$	0	0
$311$	3.12221e7i	1.03796i	0.854786	+	0.518981i	$0.173688\pi$
$311$	3.12221e7i	1.03796i	−0.854786	+	0.518981i	$0.826312\pi$
$312$	0	0
$313$	− 2.24778e7i	− 0.733029i	−0.930412	−	0.366515i	$-0.880551\pi$
$313$	− 2.24778e7i	− 0.733029i	0.930412	−	0.366515i	$-0.119449\pi$
$314$	− 3.48218e7i	− 1.12477i
$315$	0	0
$316$	1.12371e6	0.0356118
$317$	2.76211e7	0.867088	0.433544	−	0.901132i	$-0.357263\pi$
$317$	2.76211e7	0.867088	0.433544	+	0.901132i	$0.357263\pi$
$318$	0	0
$319$	3.93553e7	1.21236
$320$	0	0
$321$	0	0
$322$	− 9.24634e6i	− 0.276950i
$323$	73109.2	0.00216952
$324$	0	0
$325$	0	0
$326$	4.52942e6i	0.130734i
$327$	0	0
$328$	6.70541e6i	0.190022i
$329$	3.71672e7i	1.04369i
$330$	0	0
$331$	−5.76138e6	−0.158870	−0.0794352	−	0.996840i	$-0.525312\pi$
$331$	−5.76138e6	−0.158870	−0.0794352	+	0.996840i	$0.525312\pi$
$332$	−351359.	−0.00960144
$333$	0	0
$334$	2.71793e7	0.729456
$335$	0	0
$336$	0	0
$337$	− 4.01052e7i	− 1.04788i	−0.851756	−	0.523939i	$-0.824462\pi$
$337$	− 4.01052e7i	− 1.04788i	0.851756	−	0.523939i	$-0.175538\pi$
$338$	3.68635e7	0.954656
$339$	0	0
$340$	0	0
$341$	− 5.33535e7i	− 1.34555i
$342$	0	0
$343$	− 504328.i	− 0.0124977i
$344$	687873.i	0.0168979i
$345$	0	0
$346$	2.01945e7	0.487535
$347$	−6.78127e7	−1.62302	−0.811508	−	0.584341i	$-0.801353\pi$
$347$	−6.78127e7	−1.62302	−0.811508	+	0.584341i	$0.801353\pi$
$348$	0	0
$349$	4.20638e7	0.989538	0.494769	−	0.869024i	$-0.335253\pi$
$349$	4.20638e7	0.989538	0.494769	+	0.869024i	$0.335253\pi$
$350$	0	0
$351$	0	0
$352$	− 7.76602e6i	− 0.178062i
$353$	−1.75976e7	−0.400063	−0.200032	−	0.979789i	$-0.564104\pi$
$353$	−1.75976e7	−0.400063	−0.200032	+	0.979789i	$0.564104\pi$
$354$	0	0
$355$	0	0
$356$	4.13851e6i	0.0917263i
$357$	0	0
$358$	− 4.20699e7i	− 0.916901i
$359$	1.39920e7i	0.302410i	0.988502	+	0.151205i	$0.0483154\pi$
$359$	1.39920e7i	0.302410i	−0.988502	+	0.151205i	$0.951685\pi$
$360$	0	0
$361$	−1.40523e7	−0.298694
$362$	5.87249e7	1.23793
$363$	0	0
$364$	−5.21636e7	−1.08159
$365$	0	0
$366$	0	0
$367$	− 2.65855e7i	− 0.537832i	−0.963164	−	0.268916i	$-0.913335\pi$
$367$	− 2.65855e7i	− 0.537832i	0.963164	−	0.268916i	$-0.0866653\pi$
$368$	−3.45819e6	−0.0693914
$369$	0	0
$370$	0	0
$371$	− 1.15549e8i	− 2.26279i
$372$	0	0
$373$	− 1.78829e7i	− 0.344598i	−0.985045	−	0.172299i	$-0.944881\pi$
$373$	− 1.78829e7i	− 0.344598i	0.985045	−	0.172299i	$-0.0551195\pi$
$374$	96528.6i	0.00184519i
$375$	0	0
$376$	1.39008e7	0.261503
$377$	−9.88671e7	−1.84513
$378$	0	0
$379$	−7.20978e7	−1.32435	−0.662177	−	0.749347i	$-0.730367\pi$
$379$	−7.20978e7	−1.32435	−0.662177	+	0.749347i	$0.730367\pi$
$380$	0	0
$381$	0	0
$382$	− 7.35055e7i	− 1.31865i
$383$	8.68648e6	0.154614	0.0773068	−	0.997007i	$-0.475368\pi$
$383$	8.68648e6	0.154614	0.0773068	+	0.997007i	$0.475368\pi$
$384$	0	0
$385$	0	0
$386$	− 2.22425e7i	− 0.386742i
$387$	0	0
$388$	− 1.02856e7i	− 0.176089i
$389$	4.94411e7i	0.839923i	0.907542	+	0.419962i	$0.137956\pi$
$389$	4.94411e7i	0.839923i	−0.907542	+	0.419962i	$0.862044\pi$
$390$	0	0
$391$	42984.0	0.000719079 0
$392$	2.11081e7	0.350422
$393$	0	0
$394$	3.04320e7	0.497556
$395$	0	0
$396$	0	0
$397$	1.56911e7i	0.250774i	0.992108	+	0.125387i	$0.0400172\pi$
$397$	1.56911e7i	0.250774i	−0.992108	+	0.125387i	$0.959983\pi$
$398$	−3.20121e6	−0.0507768
$399$	0	0
$400$	0	0
$401$	− 4.74514e7i	− 0.735895i	−0.929847	−	0.367947i	$-0.880061\pi$
$401$	− 4.74514e7i	− 0.735895i	0.929847	−	0.367947i	$-0.119939\pi$
$402$	0	0
$403$	1.34033e8i	2.04784i
$404$	2.14010e7i	0.324556i
$405$	0	0
$406$	8.03711e7	1.20094
$407$	−7.04203e7	−1.04451
$408$	0	0
$409$	1.15512e8	1.68832	0.844162	−	0.536088i	$-0.180099\pi$
$409$	1.15512e8	1.68832	0.844162	+	0.536088i	$0.180099\pi$
$410$	0	0
$411$	0	0
$412$	− 6.37892e7i	− 0.912127i
$413$	1.20923e8	1.71656
$414$	0	0
$415$	0	0
$416$	1.95095e7i	0.270999i
$417$	0	0
$418$	4.35625e7i	0.596464i
$419$	− 1.46693e8i	− 1.99420i	−0.0761306	−	0.997098i	$-0.524257\pi$
$419$	− 1.46693e8i	− 1.99420i	0.0761306	−	0.997098i	$-0.475743\pi$
$420$	0	0
$421$	1.39239e8	1.86601	0.933005	−	0.359863i	$-0.117176\pi$
$421$	1.39239e8	1.86601	0.933005	+	0.359863i	$0.117176\pi$
$422$	7.64609e7	1.01742
$423$	0	0
$424$	−4.32161e7	−0.566955
$425$	0	0
$426$	0	0
$427$	− 6.41300e6i	− 0.0823716i
$428$	8.34137e6	0.106391
$429$	0	0
$430$	0	0
$431$	− 1.00392e8i	− 1.25391i	−0.779056	−	0.626954i	$-0.784301\pi$
$431$	− 1.00392e8i	− 1.25391i	0.779056	−	0.626954i	$-0.215699\pi$
$432$	0	0
$433$	4.00631e7i	0.493493i	0.969080	+	0.246747i	$0.0793616\pi$
$433$	4.00631e7i	0.493493i	−0.969080	+	0.246747i	$0.920638\pi$
$434$	− 1.08958e8i	− 1.33288i
$435$	0	0
$436$	−6.22259e6	−0.0750779
$437$	1.93983e7	0.232445
$438$	0	0
$439$	1.38592e8	1.63811	0.819057	−	0.573712i	$-0.194497\pi$
$439$	1.38592e8	1.63811	0.819057	+	0.573712i	$0.194497\pi$
$440$	0	0
$441$	0	0
$442$	− 242496.i	− 0.00280826i
$443$	−1.11443e8	−1.28186	−0.640929	−	0.767600i	$-0.721451\pi$
$443$	−1.11443e8	−1.28186	−0.640929	+	0.767600i	$0.721451\pi$
$444$	0	0
$445$	0	0
$446$	− 3.02915e7i	− 0.341442i
$447$	0	0
$448$	− 1.58597e7i	− 0.176385i
$449$	6.11166e7i	0.675181i	0.941293	+	0.337591i	$0.109612\pi$
$449$	6.11166e7i	0.675181i	−0.941293	+	0.337591i	$0.890388\pi$
$450$	0	0
$451$	4.96619e7	0.541370
$452$	−2.63007e7	−0.284808
$453$	0	0
$454$	−7.69691e7	−0.822524
$455$	0	0
$456$	0	0
$457$	3.56665e7i	0.373690i	0.982389	+	0.186845i	$0.0598262\pi$
$457$	3.56665e7i	0.373690i	−0.982389	+	0.186845i	$0.940174\pi$
$458$	2.45870e7	0.255923
$459$	0	0
$460$	0	0
$461$	1.51983e8i	1.55128i	0.631173	+	0.775642i	$0.282574\pi$
$461$	1.51983e8i	1.55128i	−0.631173	+	0.775642i	$0.717426\pi$
$462$	0	0
$463$	− 1.14978e8i	− 1.15844i	−0.815173	−	0.579218i	$-0.803358\pi$
$463$	− 1.14978e8i	− 1.15844i	0.815173	−	0.579218i	$-0.196642\pi$
$464$	− 3.00593e7i	− 0.300902i
$465$	0	0
$466$	1.14457e8	1.13106
$467$	−8.81705e7	−0.865711	−0.432855	−	0.901463i	$-0.642494\pi$
$467$	−8.81705e7	−0.865711	−0.432855	+	0.901463i	$0.642494\pi$
$468$	0	0
$469$	8.17805e7	0.792741
$470$	0	0
$471$	0	0
$472$	− 4.52260e7i	− 0.430093i
$473$	5.09456e6	0.0481419
$474$	0	0
$475$	0	0
$476$	197130.i	0.00182781i
$477$	0	0
$478$	− 1.15370e8i	− 1.05635i
$479$	8.94388e7i	0.813803i	0.913472	+	0.406902i	$0.133391\pi$
$479$	8.94388e7i	0.813803i	−0.913472	+	0.406902i	$0.866609\pi$
$480$	0	0
$481$	1.76908e8	1.58969
$482$	1.76546e7	0.157659
$483$	0	0
$484$	−827104.	−0.00729498
$485$	0	0
$486$	0	0
$487$	− 7.51688e7i	− 0.650805i	−0.945576	−	0.325403i	$-0.894500\pi$
$487$	− 7.51688e7i	− 0.650805i	0.945576	−	0.325403i	$-0.105500\pi$
$488$	−2.39851e6	−0.0206387
$489$	0	0
$490$	0	0
$491$	4.50822e7i	0.380856i	0.981701	+	0.190428i	$0.0609876\pi$
$491$	4.50822e7i	0.380856i	−0.981701	+	0.190428i	$0.939012\pi$
$492$	0	0
$493$	373626.i	0.00311815i
$494$	− 1.09436e8i	− 0.907780i
$495$	0	0
$496$	−4.07511e7	−0.333960
$497$	−2.57230e8	−2.09533
$498$	0	0
$499$	−9.15458e7	−0.736778	−0.368389	−	0.929672i	$-0.620091\pi$
$499$	−9.15458e7	−0.736778	−0.368389	+	0.929672i	$0.620091\pi$
$500$	0	0
$501$	0	0
$502$	− 2.88228e7i	− 0.227837i
$503$	−1.61043e8	−1.26543	−0.632713	−	0.774386i	$-0.718059\pi$
$503$	−1.61043e8	−1.26543	−0.632713	+	0.774386i	$0.718059\pi$
$504$	0	0
$505$	0	0
$506$	2.56122e7i	0.197695i
$507$	0	0
$508$	9.78308e7i	0.746250i
$509$	− 2.39995e7i	− 0.181990i	−0.995851	−	0.0909951i	$-0.970995\pi$
$509$	− 2.39995e7i	− 0.181990i	0.995851	−	0.0909951i	$-0.0290048\pi$
$510$	0	0
$511$	−1.14294e8	−0.856564
$512$	−5.93164e6	−0.0441942
$513$	0	0
$514$	8.16702e7	0.601415
$515$	0	0
$516$	0	0
$517$	− 1.02953e8i	− 0.745018i
$518$	−1.43812e8	−1.03468
$519$	0	0
$520$	0	0
$521$	− 9.00897e7i	− 0.637033i	−0.947917	−	0.318517i	$-0.896815\pi$
$521$	− 9.00897e7i	− 0.637033i	0.947917	−	0.318517i	$-0.103185\pi$
$522$	0	0
$523$	3.77691e7i	0.264016i	0.991249	+	0.132008i	$0.0421425\pi$
$523$	3.77691e7i	0.264016i	−0.991249	+	0.132008i	$0.957857\pi$
$524$	9.83641e7i	0.683664i
$525$	0	0
$526$	−1.76815e8	−1.21496
$527$	506520.	0.00346071
$528$	0	0
$529$	−1.36631e8	−0.922957
$530$	0	0
$531$	0	0
$532$	8.89631e7i	0.590847i
$533$	−1.24759e8	−0.823931
$534$	0	0
$535$	0	0
$536$	− 3.05865e7i	− 0.198626i
$537$	0	0
$538$	1.42178e6i	0.00913034i
$539$	− 1.56332e8i	− 0.998347i
$540$	0	0
$541$	2.54800e7	0.160919	0.0804595	−	0.996758i	$-0.474361\pi$
$541$	2.54800e7	0.160919	0.0804595	+	0.996758i	$0.474361\pi$
$542$	−1.67669e8	−1.05306
$543$	0	0
$544$	73728.0	0.000457969 0
$545$	0	0
$546$	0	0
$547$	2.05216e8i	1.25386i	0.779076	+	0.626930i	$0.215689\pi$
$547$	2.05216e8i	1.25386i	−0.779076	+	0.626930i	$0.784311\pi$
$548$	1.43492e8	0.871938
$549$	0	0
$550$	0	0
$551$	1.68614e8i	1.00795i
$552$	0	0
$553$	− 1.69961e7i	− 0.100502i
$554$	− 7.47908e7i	− 0.439865i
$555$	0	0
$556$	3.49545e7	0.203366
$557$	2.41143e8	1.39543	0.697715	−	0.716375i	$-0.254200\pi$
$557$	2.41143e8	1.39543	0.697715	+	0.716375i	$0.254200\pi$
$558$	0	0
$559$	−1.27984e7	−0.0732690
$560$	0	0
$561$	0	0
$562$	3.46403e7i	0.195152i
$563$	−1.68877e8	−0.946337	−0.473168	−	0.880972i	$-0.656890\pi$
$563$	−1.68877e8	−0.946337	−0.473168	+	0.880972i	$0.656890\pi$
$564$	0	0
$565$	0	0
$566$	− 3.81456e7i	− 0.210375i
$567$	0	0
$568$	9.62058e7i	0.524996i
$569$	2.43995e8i	1.32448i	0.749293	+	0.662238i	$0.230393\pi$
$569$	2.43995e8i	1.32448i	−0.749293	+	0.662238i	$0.769607\pi$
$570$	0	0
$571$	2.41502e8	1.29722	0.648608	−	0.761123i	$-0.275352\pi$
$571$	2.41502e8	1.29722	0.648608	+	0.761123i	$0.275352\pi$
$572$	1.44493e8	0.772071
$573$	0	0
$574$	1.01419e8	0.536272
$575$	0	0
$576$	0	0
$577$	− 4.93979e7i	− 0.257147i	−0.991700	−	0.128573i	$-0.958960\pi$
$577$	− 4.93979e7i	− 0.257147i	0.991700	−	0.128573i	$-0.0410398\pi$
$578$	1.36542e8	0.707102
$579$	0	0
$580$	0	0
$581$	5.31430e6i	0.0270968i
$582$	0	0
$583$	3.20069e8i	1.61525i
$584$	4.27466e7i	0.214617i
$585$	0	0
$586$	5.67038e7	0.281786
$587$	1.72052e8	0.850639	0.425320	−	0.905043i	$-0.360162\pi$
$587$	1.72052e8	0.850639	0.425320	+	0.905043i	$0.360162\pi$
$588$	0	0
$589$	2.28588e8	1.11869
$590$	0	0
$591$	0	0
$592$	5.37866e7i	0.259244i
$593$	2.70643e8	1.29788	0.648938	−	0.760841i	$-0.275213\pi$
$593$	2.70643e8	1.29788	0.648938	+	0.760841i	$0.275213\pi$
$594$	0	0
$595$	0	0
$596$	7.10680e7i	0.335687i
$597$	0	0
$598$	− 6.43423e7i	− 0.300880i
$599$	1.73299e8i	0.806337i	0.915126	+	0.403169i	$0.132091\pi$
$599$	1.73299e8i	0.806337i	−0.915126	+	0.403169i	$0.867909\pi$
$600$	0	0
$601$	−4.31090e8	−1.98584	−0.992921	−	0.118775i	$-0.962103\pi$
$601$	−4.31090e8	−1.98584	−0.992921	+	0.118775i	$0.962103\pi$
$602$	1.04041e7	0.0476886
$603$	0	0
$604$	−1.30519e8	−0.592327
$605$	0	0
$606$	0	0
$607$	1.66991e7i	0.0746665i	0.999303	+	0.0373332i	$0.0118863\pi$
$607$	1.66991e7i	0.0746665i	−0.999303	+	0.0373332i	$0.988114\pi$
$608$	3.32728e7	0.148040
$609$	0	0
$610$	0	0
$611$	2.58635e8i	1.13387i
$612$	0	0
$613$	1.92321e8i	0.834920i	0.908695	+	0.417460i	$0.137080\pi$
$613$	1.92321e8i	0.834920i	−0.908695	+	0.417460i	$0.862920\pi$
$614$	2.95887e7i	0.127827i
$615$	0	0
$616$	−1.17461e8	−0.502518
$617$	1.87023e8	0.796233	0.398117	−	0.917335i	$-0.369664\pi$
$617$	1.87023e8	0.796233	0.398117	+	0.917335i	$0.369664\pi$
$618$	0	0
$619$	−2.54873e8	−1.07461	−0.537307	−	0.843387i	$-0.680558\pi$
$619$	−2.54873e8	−1.07461	−0.537307	+	0.843387i	$0.680558\pi$
$620$	0	0
$621$	0	0
$622$	− 1.76619e8i	− 0.733950i
$623$	6.25950e7	0.258866
$624$	0	0
$625$	0	0
$626$	1.27154e8i	0.518330i
$627$	0	0
$628$	1.96982e8i	0.795329i
$629$	− 668547.i	− 0.00268646i
$630$	0	0
$631$	9.23602e7	0.367618	0.183809	−	0.982962i	$-0.441157\pi$
$631$	9.23602e7	0.367618	0.183809	+	0.982962i	$0.441157\pi$
$632$	−6.35668e6	−0.0251813
$633$	0	0
$634$	−1.56249e8	−0.613124
$635$	0	0
$636$	0	0
$637$	3.92732e8i	1.51942i
$638$	−2.22627e8	−0.857267
$639$	0	0
$640$	0	0
$641$	− 4.24666e8i	− 1.61240i	−0.591643	−	0.806200i	$-0.701520\pi$
$641$	− 4.24666e8i	− 1.61240i	0.591643	−	0.806200i	$-0.298480\pi$
$642$	0	0
$643$	− 3.75946e8i	− 1.41414i	−0.707143	−	0.707071i	$-0.750016\pi$
$643$	− 3.75946e8i	− 1.41414i	0.707143	−	0.707071i	$-0.249984\pi$
$644$	5.23052e7i	0.195834i
$645$	0	0
$646$	−413568.	−0.00153408
$647$	−2.63747e7	−0.0973813	−0.0486906	−	0.998814i	$-0.515505\pi$
$647$	−2.63747e7	−0.0973813	−0.0486906	+	0.998814i	$0.515505\pi$
$648$	0	0
$649$	−3.34955e8	−1.22533
$650$	0	0
$651$	0	0
$652$	− 2.56223e7i	− 0.0924432i
$653$	2.58756e8	0.929291	0.464645	−	0.885497i	$-0.346182\pi$
$653$	2.58756e8	0.929291	0.464645	+	0.885497i	$0.346182\pi$
$654$	0	0
$655$	0	0
$656$	− 3.79315e7i	− 0.134366i
$657$	0	0
$658$	− 2.10250e8i	− 0.738002i
$659$	1.39345e8i	0.486895i	0.969914	+	0.243447i	$0.0782783\pi$
$659$	1.39345e8i	0.486895i	−0.969914	+	0.243447i	$0.921722\pi$
$660$	0	0
$661$	−4.72545e8	−1.63621	−0.818104	−	0.575070i	$-0.804975\pi$
$661$	−4.72545e8	−1.63621	−0.818104	+	0.575070i	$0.804975\pi$
$662$	3.25913e7	0.112338
$663$	0	0
$664$	1.98758e6	0.00678924
$665$	0	0
$666$	0	0
$667$	9.91354e7i	0.334081i
$668$	−1.53749e8	−0.515804
$669$	0	0
$670$	0	0
$671$	1.77639e7i	0.0587992i
$672$	0	0
$673$	− 5.48833e8i	− 1.80051i	−0.435364	−	0.900254i	$-0.643380\pi$
$673$	− 5.48833e8i	− 1.80051i	0.435364	−	0.900254i	$-0.356620\pi$
$674$	2.26869e8i	0.740961i
$675$	0	0
$676$	−2.08532e8	−0.675044
$677$	1.00760e8	0.324731	0.162365	−	0.986731i	$-0.448088\pi$
$677$	1.00760e8	0.324731	0.162365	+	0.986731i	$0.448088\pi$
$678$	0	0
$679$	−1.55569e8	−0.496952
$680$	0	0
$681$	0	0
$682$	3.01813e8i	0.951447i
$683$	−313056.	−0.000982562 0	−0.000491281	−	1.00000i	$-0.500156\pi$
$683$	−313056.	−0.000982562 0	−0.000491281		1.00000i	$0.500156\pi$
$684$	0	0
$685$	0	0
$686$	2.85291e6i	0.00883722i
$687$	0	0
$688$	− 3.89120e6i	− 0.0119486i
$689$	− 8.04068e8i	− 2.45830i
$690$	0	0
$691$	−3.72812e8	−1.12994	−0.564971	−	0.825111i	$-0.691113\pi$
$691$	−3.72812e8	−1.12994	−0.564971	+	0.825111i	$0.691113\pi$
$692$	−1.14238e8	−0.344739
$693$	0	0
$694$	3.83607e8	1.14765
$695$	0	0
$696$	0	0
$697$	471474.i	0.00139239i
$698$	−2.37949e8	−0.699709
$699$	0	0
$700$	0	0
$701$	6.21170e8i	1.80325i	0.432517	+	0.901626i	$0.357626\pi$
$701$	6.21170e8i	1.80325i	−0.432517	+	0.901626i	$0.642374\pi$
$702$	0	0
$703$	− 3.01709e8i	− 0.868406i
$704$	4.39312e7i	0.125909i
$705$	0	0
$706$	9.95469e7	0.282887
$707$	3.23690e8	0.915947
$708$	0	0
$709$	2.46510e8	0.691666	0.345833	−	0.938296i	$-0.387596\pi$
$709$	2.46510e8	0.691666	0.345833	+	0.938296i	$0.387596\pi$
$710$	0	0
$711$	0	0
$712$	− 2.34109e7i	− 0.0648603i
$713$	1.34397e8	0.370783
$714$	0	0
$715$	0	0
$716$	2.37983e8i	0.648347i
$717$	0	0
$718$	− 7.91508e7i	− 0.213836i
$719$	9.60389e7i	0.258381i	0.991620	+	0.129191i	$0.0412379\pi$
$719$	9.60389e7i	0.258381i	−0.991620	+	0.129191i	$0.958762\pi$
$720$	0	0
$721$	−9.64811e8	−2.57417
$722$	7.94921e7	0.211209
$723$	0	0
$724$	−3.32198e8	−0.875350
$725$	0	0
$726$	0	0
$727$	3.91371e8i	1.01856i	0.860602	+	0.509278i	$0.170088\pi$
$727$	3.91371e8i	1.01856i	−0.860602	+	0.509278i	$0.829912\pi$
$728$	2.95082e8	0.764801
$729$	0	0
$730$	0	0
$731$	48366.1i	0 0.000123819i
$732$	0	0
$733$	− 3.49078e7i	− 0.0886361i	−0.999017	−	0.0443181i	$-0.985889\pi$
$733$	− 3.49078e7i	− 0.0886361i	0.999017	−	0.0443181i	$-0.0141115\pi$
$734$	1.50390e8i	0.380304i
$735$	0	0
$736$	1.95625e7	0.0490671
$737$	−2.26531e8	−0.565881
$738$	0	0
$739$	3.02999e8	0.750773	0.375386	−	0.926868i	$-0.377510\pi$
$739$	3.02999e8	0.750773	0.375386	+	0.926868i	$0.377510\pi$
$740$	0	0
$741$	0	0
$742$	6.53644e8i	1.60004i
$743$	2.45628e8	0.598842	0.299421	−	0.954121i	$-0.403207\pi$
$743$	2.45628e8	0.598842	0.299421	+	0.954121i	$0.403207\pi$
$744$	0	0
$745$	0	0
$746$	1.01161e8i	0.243667i
$747$	0	0
$748$	− 546048.i	− 0.00130475i
$749$	− 1.26163e8i	− 0.300253i
$750$	0	0
$751$	−8.23270e7	−0.194367	−0.0971835	−	0.995266i	$-0.530983\pi$
$751$	−8.23270e7	−0.194367	−0.0971835	+	0.995266i	$0.530983\pi$
$752$	−7.86348e7	−0.184910
$753$	0	0
$754$	5.59277e8	1.30471
$755$	0	0
$756$	0	0
$757$	− 6.03579e8i	− 1.39138i	−0.718341	−	0.695691i	$-0.755098\pi$
$757$	− 6.03579e8i	− 1.39138i	0.718341	−	0.695691i	$-0.244902\pi$
$758$	4.07847e8	0.936460
$759$	0	0
$760$	0	0
$761$	− 2.32982e8i	− 0.528651i	−0.964434	−	0.264325i	$-0.914851\pi$
$761$	− 2.32982e8i	− 0.528651i	0.964434	−	0.264325i	$-0.0851493\pi$
$762$	0	0
$763$	9.41167e7i	0.211882i
$764$	4.15810e8i	0.932426i
$765$	0	0
$766$	−4.91382e7	−0.109328
$767$	8.41463e8	1.86487
$768$	0	0
$769$	−8.15796e8	−1.79392	−0.896958	−	0.442115i	$-0.854228\pi$
$769$	−8.15796e8	−1.79392	−0.896958	+	0.442115i	$0.854228\pi$
$770$	0	0
$771$	0	0
$772$	1.25823e8i	0.273468i
$773$	−3.66587e8	−0.793667	−0.396833	−	0.917891i	$-0.629891\pi$
$773$	−3.66587e8	−0.793667	−0.396833	+	0.917891i	$0.629891\pi$
$774$	0	0
$775$	0	0
$776$	5.81840e7i	0.124514i
$777$	0	0
$778$	− 2.79681e8i	− 0.593915i
$779$	2.12772e8i	0.450093i
$780$	0	0
$781$	7.12524e8	1.49571
$782$	−243154.	−0.000508466 0
$783$	0	0
$784$	−1.19406e8	−0.247786
$785$	0	0
$786$	0	0
$787$	− 4.02462e8i	− 0.825659i	−0.910808	−	0.412830i	$-0.864540\pi$
$787$	− 4.02462e8i	− 0.825659i	0.910808	−	0.412830i	$-0.135460\pi$
$788$	−1.72150e8	−0.351825
$789$	0	0
$790$	0	0
$791$	3.97798e8i	0.803773i
$792$	0	0
$793$	− 4.46260e7i	− 0.0894887i
$794$	− 8.87623e7i	− 0.177324i
$795$	0	0
$796$	1.81088e7	0.0359046
$797$	5.18940e8	1.02504	0.512521	−	0.858675i	$-0.328712\pi$
$797$	5.18940e8	1.02504	0.512521	+	0.858675i	$0.328712\pi$
$798$	0	0
$799$	977400.	0.00191616
$800$	0	0
$801$	0	0
$802$	2.68426e8i	0.520356i
$803$	3.16592e8	0.611440
$804$	0	0
$805$	0	0
$806$	− 7.58205e8i	− 1.44804i
$807$	0	0
$808$	− 1.21062e8i	− 0.229496i
$809$	3.04036e6i	0.00574221i	0.999996	+	0.00287110i	$0.000913902\pi$
$809$	3.04036e6i	0.00574221i	−0.999996	+	0.00287110i	$0.999086\pi$
$810$	0	0
$811$	2.25521e8	0.422790	0.211395	−	0.977401i	$-0.432199\pi$
$811$	2.25521e8	0.422790	0.211395	+	0.977401i	$0.432199\pi$
$812$	−4.54648e8	−0.849194
$813$	0	0
$814$	3.98357e8	0.738583
$815$	0	0
$816$	0	0
$817$	2.18272e7i	0.0400250i
$818$	−6.53432e8	−1.19383
$819$	0	0
$820$	0	0
$821$	2.77035e8i	0.500617i	0.968166	+	0.250309i	$0.0805321\pi$
$821$	2.77035e8i	0.500617i	−0.968166	+	0.250309i	$0.919468\pi$
$822$	0	0
$823$	7.07336e8i	1.26890i	0.772965	+	0.634448i	$0.218773\pi$
$823$	7.07336e8i	1.26890i	−0.772965	+	0.634448i	$0.781227\pi$
$824$	3.60846e8i	0.644971i
$825$	0	0
$826$	−6.84043e8	−1.21379
$827$	2.66346e8	0.470900	0.235450	−	0.971886i	$-0.424344\pi$
$827$	2.66346e8	0.470900	0.235450	+	0.971886i	$0.424344\pi$
$828$	0	0
$829$	−5.03826e8	−0.884336	−0.442168	−	0.896932i	$-0.645791\pi$
$829$	−5.03826e8	−0.884336	−0.442168	+	0.896932i	$0.645791\pi$
$830$	0	0
$831$	0	0
$832$	− 1.10363e8i	− 0.191625i
$833$	1.48416e6	0.00256772
$834$	0	0
$835$	0	0
$836$	− 2.46427e8i	− 0.421763i
$837$	0	0
$838$	8.29822e8i	1.41011i
$839$	− 7.63364e8i	− 1.29255i	−0.763106	−	0.646273i	$-0.776327\pi$
$839$	− 7.63364e8i	− 1.29255i	0.763106	−	0.646273i	$-0.223673\pi$
$840$	0	0
$841$	−2.66883e8	−0.448676
$842$	−7.87654e8	−1.31947
$843$	0	0
$844$	−4.32528e8	−0.719427
$845$	0	0
$846$	0	0
$847$	1.25099e7i	0.0205876i
$848$	2.44467e8	0.400897
$849$	0	0
$850$	0	0
$851$	− 1.77388e8i	− 0.287829i
$852$	0	0
$853$	1.87985e7i	0.0302884i	0.999885	+	0.0151442i	$0.00482073\pi$
$853$	1.87985e7i	0.0302884i	−0.999885	+	0.0151442i	$0.995179\pi$
$854$	3.62774e7i	0.0582455i
$855$	0	0
$856$	−4.71859e7	−0.0752300
$857$	6.86427e8	1.09057	0.545283	−	0.838252i	$-0.316422\pi$
$857$	6.86427e8	1.09057	0.545283	+	0.838252i	$0.316422\pi$
$858$	0	0
$859$	−5.51932e8	−0.870775	−0.435387	−	0.900243i	$-0.643389\pi$
$859$	−5.51932e8	−0.870775	−0.435387	+	0.900243i	$0.643389\pi$
$860$	0	0
$861$	0	0
$862$	5.67901e8i	0.886647i
$863$	3.65665e8	0.568920	0.284460	−	0.958688i	$-0.408186\pi$
$863$	3.65665e8	0.568920	0.284460	+	0.958688i	$0.408186\pi$
$864$	0	0
$865$	0	0
$866$	− 2.26631e8i	− 0.348952i
$867$	0	0
$868$	6.16360e8i	0.942487i
$869$	4.70791e7i	0.0717413i
$870$	0	0
$871$	5.69084e8	0.861236
$872$	3.52003e7	0.0530881
$873$	0	0
$874$	−1.09733e8	−0.164363
$875$	0	0
$876$	0	0
$877$	− 5.85387e8i	− 0.867849i	−0.900949	−	0.433925i	$-0.857128\pi$
$877$	− 5.85387e8i	− 0.867849i	0.900949	−	0.433925i	$-0.142872\pi$
$878$	−7.83994e8	−1.15832
$879$	0	0
$880$	0	0
$881$	− 4.29761e8i	− 0.628491i	−0.949342	−	0.314246i	$-0.898248\pi$
$881$	− 4.29761e8i	− 0.628491i	0.949342	−	0.314246i	$-0.101752\pi$
$882$	0	0
$883$	− 2.20085e8i	− 0.319675i	−0.987143	−	0.159837i	$-0.948903\pi$
$883$	− 2.20085e8i	− 0.319675i	0.987143	−	0.159837i	$-0.0510970\pi$
$884$	1.37176e6i	0.00198574i
$885$	0	0
$886$	6.30415e8	0.906411
$887$	−1.17196e9	−1.67936	−0.839678	−	0.543084i	$-0.817256\pi$
$887$	−1.17196e9	−1.67936	−0.839678	+	0.543084i	$0.817256\pi$
$888$	0	0
$889$	1.47969e9	2.10604
$890$	0	0
$891$	0	0
$892$	1.71355e8i	0.241436i
$893$	4.41092e8	0.619405
$894$	0	0
$895$	0	0
$896$	8.97161e7i	0.124723i
$897$	0	0
$898$	− 3.45728e8i	− 0.477425i
$899$	1.16820e9i	1.60783i
$900$	0	0
$901$	−3.03863e6	−0.00415436
$902$	−2.80930e8	−0.382806
$903$	0	0
$904$	1.48779e8	0.201390
$905$	0	0
$906$	0	0
$907$	7.31614e8i	0.980529i	0.871574	+	0.490264i	$0.163100\pi$
$907$	7.31614e8i	0.980529i	−0.871574	+	0.490264i	$0.836900\pi$
$908$	4.35403e8	0.581612
$909$	0	0
$910$	0	0
$911$	9.18595e8i	1.21498i	0.794327	+	0.607490i	$0.207823\pi$
$911$	9.18595e8i	1.21498i	−0.794327	+	0.607490i	$0.792177\pi$
$912$	0	0
$913$	− 1.47205e7i	− 0.0193425i
$914$	− 2.01760e8i	− 0.264239i
$915$	0	0
$916$	−1.39085e8	−0.180965
$917$	1.48776e9	1.92941
$918$	0	0
$919$	2.15987e8	0.278279	0.139139	−	0.990273i	$-0.455566\pi$
$919$	2.15987e8	0.278279	0.139139	+	0.990273i	$0.455566\pi$
$920$	0	0
$921$	0	0
$922$	− 8.59744e8i	− 1.09692i
$923$	−1.78998e9	−2.27637
$924$	0	0
$925$	0	0
$926$	6.50414e8i	0.819138i
$927$	0	0
$928$	1.70041e8i	0.212770i
$929$	− 3.10124e8i	− 0.386802i	−0.981120	−	0.193401i	$-0.938048\pi$
$929$	− 3.10124e8i	− 0.386802i	0.981120	−	0.193401i	$-0.0619518\pi$
$930$	0	0
$931$	6.69791e8	0.830023
$932$	−6.47466e8	−0.799778
$933$	0	0
$934$	4.98768e8	0.612150
$935$	0	0
$936$	0	0
$937$	− 7.42448e8i	− 0.902501i	−0.892397	−	0.451250i	$-0.850978\pi$
$937$	− 7.42448e8i	− 0.902501i	0.892397	−	0.451250i	$-0.149022\pi$
$938$	−4.62620e8	−0.560553
$939$	0	0
$940$	0	0
$941$	1.81766e8i	0.218144i	0.994034	+	0.109072i	$0.0347879\pi$
$941$	1.81766e8i	0.218144i	−0.994034	+	0.109072i	$0.965212\pi$
$942$	0	0
$943$	1.25098e8i	0.149181i
$944$	2.55837e8i	0.304121i
$945$	0	0
$946$	−2.88192e7	−0.0340415
$947$	−8.59189e8	−1.01167	−0.505835	−	0.862630i	$-0.668816\pi$
$947$	−8.59189e8	−1.01167	−0.505835	+	0.862630i	$0.668816\pi$
$948$	0	0
$949$	−7.95333e8	−0.930573
$950$	0	0
$951$	0	0
$952$	− 1.11514e6i	− 0.00129246i
$953$	6.86819e8	0.793530	0.396765	−	0.917920i	$-0.370133\pi$
$953$	6.86819e8	0.793530	0.396765	+	0.917920i	$0.370133\pi$
$954$	0	0
$955$	0	0
$956$	6.52630e8i	0.746953i
$957$	0	0
$958$	− 5.05942e8i	− 0.575446i
$959$	− 2.17031e9i	− 2.46075i
$960$	0	0
$961$	6.96218e8	0.784468
$962$	−1.00074e9	−1.12408
$963$	0	0
$964$	−9.98697e7	−0.111481
$965$	0	0
$966$	0	0
$967$	− 1.09411e9i	− 1.20999i	−0.796230	−	0.604995i	$-0.793175\pi$
$967$	− 1.09411e9i	− 1.20999i	0.796230	−	0.604995i	$-0.206825\pi$
$968$	4.67881e6	0.00515833
$969$	0	0
$970$	0	0
$971$	4.43115e8i	0.484014i	0.970274	+	0.242007i	$0.0778058\pi$
$971$	4.43115e8i	0.484014i	−0.970274	+	0.242007i	$0.922194\pi$
$972$	0	0
$973$	− 5.28687e8i	− 0.573931i
$974$	4.25219e8i	0.460189i
$975$	0	0
$976$	1.35680e7	0.0145937
$977$	−1.19004e9	−1.27608	−0.638042	−	0.770001i	$-0.720256\pi$
$977$	−1.19004e9	−1.27608	−0.638042	+	0.770001i	$0.720256\pi$
$978$	0	0
$979$	−1.73387e8	−0.184786
$980$	0	0
$981$	0	0
$982$	− 2.55024e8i	− 0.269306i
$983$	−1.18187e9	−1.24425	−0.622125	−	0.782918i	$-0.713730\pi$
$983$	−1.18187e9	−1.24425	−0.622125	+	0.782918i	$0.713730\pi$
$984$	0	0
$985$	0	0
$986$	− 2.11355e6i	− 0.00220486i
$987$	0	0
$988$	6.19065e8i	0.641897i
$989$	1.28331e7i	0.0132661i
$990$	0	0
$991$	5.09602e8	0.523613	0.261806	−	0.965120i	$-0.415682\pi$
$991$	5.09602e8	0.523613	0.261806	+	0.965120i	$0.415682\pi$
$992$	2.30523e8	0.236145
$993$	0	0
$994$	1.45511e9	1.48162
$995$	0	0
$996$	0	0
$997$	− 9.90780e8i	− 0.999751i	−0.866097	−	0.499875i	$-0.833379\pi$
$997$	− 9.90780e8i	− 0.999751i	0.866097	−	0.499875i	$-0.166621\pi$
$998$	5.17861e8	0.520981
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.7.b.a.449.1		4
3.2	odd	2	inner	450.7.b.a.449.3		4
5.2	odd	4		450.7.d.a.251.1		2
5.3	odd	4		18.7.b.a.17.2	yes	2
5.4	even	2	inner	450.7.b.a.449.4		4
15.2	even	4		450.7.d.a.251.2		2
15.8	even	4		18.7.b.a.17.1	✓	2
15.14	odd	2	inner	450.7.b.a.449.2		4
20.3	even	4		144.7.e.d.17.2		2
40.3	even	4		576.7.e.k.449.1		2
40.13	odd	4		576.7.e.b.449.1		2
45.13	odd	12		162.7.d.d.107.1		4
45.23	even	12		162.7.d.d.107.2		4
45.38	even	12		162.7.d.d.53.1		4
45.43	odd	12		162.7.d.d.53.2		4
60.23	odd	4		144.7.e.d.17.1		2
120.53	even	4		576.7.e.b.449.2		2
120.83	odd	4		576.7.e.k.449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.7.b.a.17.1	✓	2	15.8	even	4
18.7.b.a.17.2	yes	2	5.3	odd	4
144.7.e.d.17.1		2	60.23	odd	4
144.7.e.d.17.2		2	20.3	even	4
162.7.d.d.53.1		4	45.38	even	12
162.7.d.d.53.2		4	45.43	odd	12
162.7.d.d.107.1		4	45.13	odd	12
162.7.d.d.107.2		4	45.23	even	12
450.7.b.a.449.1		4	1.1	even	1	trivial
450.7.b.a.449.2		4	15.14	odd	2	inner
450.7.b.a.449.3		4	3.2	odd	2	inner
450.7.b.a.449.4		4	5.4	even	2	inner
450.7.d.a.251.1		2	5.2	odd	4
450.7.d.a.251.2		2	15.2	even	4
576.7.e.b.449.1		2	40.13	odd	4
576.7.e.b.449.2		2	120.53	even	4
576.7.e.k.449.1		2	40.3	even	4
576.7.e.k.449.2		2	120.83	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	450.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.65685 q^{2} +32.0000 q^{4} -484.000i q^{7} -181.019 q^{8} +1340.67i q^{11} -3368.00i q^{13} +2737.92i q^{14} +1024.00 q^{16} -12.7279 q^{17} -5744.00 q^{19} -7584.00i q^{22} -3377.14 q^{23} +19052.3i q^{26} -15488.0i q^{28} -29354.8i q^{29} -39796.0 q^{31} -5792.62 q^{32} +72.0000 q^{34} +52526.0i q^{37} +32493.0 q^{38} -37042.5i q^{41} -3800.00i q^{43} +42901.6i q^{44} +19104.0 q^{46} -76791.8 q^{47} -116607. q^{49} -107776. i q^{52} +238738. q^{53} +87613.4i q^{56} +166056. i q^{58} +249841. i q^{59} +13250.0 q^{61} +225120. q^{62} +32768.0 q^{64} +168968. i q^{67} -407.294 q^{68} -531467. i q^{71} -236144. i q^{73} -297132. i q^{74} -183808. q^{76} +648886. q^{77} +35116.0 q^{79} +209544. i q^{82} -10980.0 q^{83} +21496.0i q^{86} -242688. i q^{88} +129328. i q^{89} -1.63011e6 q^{91} -108069. q^{92} +434400. q^{94} -321424. i q^{97} +659629. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 128 q^{4} + 4096 q^{16} - 22976 q^{19} - 159184 q^{31} + 288 q^{34} + 76416 q^{46} - 466428 q^{49} + 53000 q^{61} + 131072 q^{64} - 735232 q^{76} + 140464 q^{79} - 6520448 q^{91} + 1737600 q^{94}+O(q^{100}) \) 4 * q + 128 * q^4 + 4096 * q^16 - 22976 * q^19 - 159184 * q^31 + 288 * q^34 + 76416 * q^46 - 466428 * q^49 + 53000 * q^61 + 131072 * q^64 - 735232 * q^76 + 140464 * q^79 - 6520448 * q^91 + 1737600 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more